Dark matter Sommerfeld-enhanced annihilation and bound ...

Dark matter Sommerfeld-enhanced annihilationand bound-state decay at finite temperature

Tobias Binder,1,* Laura Covi,1,† and Kyohei Mukaida2,‡1Institute for Theoretical Physics, Georg-August University Göttingen,

Friedrich-Hund-Platz 1, Göttingen, D-37077 Germany2Deutsches Elektronen-Synchrotron (DESY), Notkestraße 85, Hamburg, D-22607 Germany

(Received 5 September 2018; published 17 December 2018)

Traditional computations of the dark matter (DM) relic abundance, for models where attractive self-interactions are mediated by light force-carriers and bound states exist, rely on the solution of a coupledsystem of classical on-shell Boltzmann equations. This idealized description misses important thermaleffects caused by the tight coupling among force-carriers and other charged relativistic species potentiallypresent during the chemical decoupling process. We develop for the first time a comprehensive ab initioderivation for the description of DM long-range interactions in the presence of a hot and dense plasmabackground directly from nonequilibrium quantum field theory. Our results clarify a few conceptionalaspects of the derivation and show that under certain conditions the finite temperature effects can lead tosizable modifications of DM Sommerfeld-enhanced annihilation and bound-state decay rates. In particular,the scattering and bound states get strongly mixed in the thermal plasma environment, representing acharacteristic difference from a pure vacuum theory computation. The main result of this work is a noveldifferential equation for the DM number density, written down in a form which is manifestly independentunder the choice of what one would interpret as a bound or a scattering state at finite temperature. Thecollision term, unifying the description of annihilation and bound-state decay, turns out to have in general anonquadratic dependence on the DM number density. This generalizes the form of the conventionalLee-Weinberg equation which is typically adopted to describe the freeze-out process. We prove that ournumber density equation is consistent with previous literature results under certain limits. In the limit ofvanishing finite temperature corrections our central equation is fully compatible with the classical on-shellBoltzmann equation treatment. So far, finite temperature corrected annihilation rates for long-range forcesystems have been estimated from a method relying on linear response theory. We prove consistencybetween the latter and our method in the linear regime close to chemical equilibrium.

DOI: 10.1103/PhysRevD.98.115023

I. INTRODUCTION

The cosmological standard model successfully describesthe evolution of the large-scale structure of our Universe. Itrequires the existence of a cold and collisionless mattercomponent, called dark matter (DM), which dominatesover the baryon content in the matter dominated era of ourUniverse. The Planck satellite measurements of the cosmicmicrowave background (CMB) temperature anisotropieshave nowadays determined the amount of dark matterto an unprecedented precision, reaching the level of

subpercentage accuracy in the observational determinationof the abundance when combining CMB and external data[1,2], e.g., measurements of the baryon acoustic oscillation.Interestingly, astrophysical observation and structure

formation on subgalactic scales might point toward thenature of dark matter as velocity-dependent self-interactingelementary particles. On the one hand, observations ofgalaxy cluster systems, where typical rotational velocitiesare of the order v0 ∼ 1000 km=s, set the most stringentbounds on the self-scattering cross section to be less thanσ=mDM ≲ 0.7ð0.1Þ cm2=g in the bullet cluster [3] (in orderto guarantee the production of elliptical halos [4,5]). On theother hand, a DM self-scattering cross section of the orderσ=mDM ∼ 1 cm2=g on dwarf-galactic scales, where veloc-ities are of the order v0 ∼ 30–100 km=s, would lead to acompelling solution of the cusp-core and diversity problemwithout strongly relying on uncertain assumptions ofmodeling the barionic feedbacks in simulations. Thisvelocity dependence of the self-scattering cross sectioncan naturally be realized in models where a light mediator

*[email protected]†[email protected]‡[email protected]

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI. Funded by SCOAP3.

PHYSICAL REVIEW D 98, 115023 (2018)

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acts as a long-range force between the dark matter particles.For a recent review on self-interacting DM, see [6].Generically, long-range forces can lead to sizable mod-

ificationsof theDMtree-level annihilation cross section in theregimewhere the annihilating particles are slow. For the mostappealing DM candidates, known as weakly interactingmassive particles (WIMP) dark matter [7–9], such that therelic abundance in the early Universe is set by the thermalfreeze-out mechanism when the DM is nonrelativistic, theseeffects can be sizable already at the time of chemicaldecoupling. Then the inclusion of the long-range forcemodification in the computation of the relic abundance isnecessary to reach the required level of the accuracy set by thePlanck precision measurement [1,2]. If the light mediatorsinduce an attractive force between the annihilating DMparticles, the total cross section is typically enhanced[10,11]which is often referred to as the Sommerfeld enhance-ment [12] or Sakharov enhancement [13]. Additionally, suchattractive forces can lead to the existence of DM bound-statesolutions [14–16]. This opens the possibility for conversionprocesses between scattering and bound states via radiativeprocesses, influencing the evolution of the abundance of thestable scattering states during the DM thermal history. DMscenarioswithSommerfeld enhancementorwithboundstateshave been widely studied in the literature [17–59] and it hasbeen shown that the main effect of such corrections is to shiftin the parameter space the upper bounds on the DM mass,otherwise the theoretically predicted DM density would betoo large (overclosure bound).Classic WIMP candidates with large corrections via

Sommerfeld enhancement or bound states are particlescharged under the electroweak interactions, like the Winoneutralino in supersymmetricmodels [60] or the first Kaluza-Klein excitation of the gauge boson in models with extradimensions [61]. For the supersymmetric case it was realizedvery early on by [10,11] that the Sommerfeld effect reducestheWino density up to 30% and pushes themass of theWinodark matter candidate to few TeVs in order to obtain thecorrect relic density. These studies have later been extendedto the case of general components of the neutralino [29,30].Similar and even stronger effects from the Sommerfeldenhancement and bound states were found in the case ofcoannihilation of theWIMP particle with charged or coloredstates [31–41]. If the electroweakly charged dark matter issufficiently heavy, the Sommerfeld enhancement or thepresence of bound states due to the exchange of electroweakgauge orHiggs bosons, see e.g., [42,43], are very generic as itwas shown e.g., in the minimal dark matter model [40,44]and in Higgs portal models [45]. In these cases, long-rangeforce effects play an important role also for the indirectdetection limits [14,15,46–48] and especially for the Winothe Sommerfeld enhancement has lead to the exclusion ofmost parameter space [49–52]. Note that this effect can beimportant alsowhen the dark matter is not itself aWIMP, butit is produced byWIMP decay out of equilibrium, like in thesuper-WIMP mechanism [62,63]. Indeed in such a scenario,

the DM inherits part of the energy density of the motherparticle and so any change in the latter freeze-out density isdirectly transferred to the superweakly interacting DM andcan relax the BBN constraints on the mother particle [53].While a lot of effort has been made to compute quantita-

tively the effects of a long-range force on theDM relic densityemploying the classical Boltzmann equationmethod, it is stillunclear if that is a sufficient description. Indeed, consideringthepresence of a thermal plasma on the long-range force leadson one side to a possible screening by the presence of athermal mass, or on the other to the issue if the coherence inthe (in principle infinite) ladder diagram exchanges betweenthe two slowly moving annihilating particles is guaranteed.Moreover, from a conceptional point of view, there is yet noconsistent formulation in the existing literature of how to dealwith long-range forces at finite temperature, especially if thedark matter is, or, enters an out-of-equilibrium state (alreadythe standard freeze-out scenario is a transition from chemicalequilibrium to out-of-chemical equilibrium). The main con-cern of our work will be to clarify conceptional aspects of thederivation and the solution of the number density equation forDM particles with attractive long-range force interactions inthe presence of a hot and dense plasma background, startingfrom first principles. From thebeginning,wework in the real-time formalism, which has a smooth connection to genericout-of-equilibrium phenomena.The simplifiedDMsystemwewould like to describe in the

presence of a thermal environment is similar to heavy quarksin a hot quark gluon plasma. For this setup it has been shownthat finite temperature effects can lead to a melting of theheavy-quark bound states which influences, e.g., the anni-hilation rate of the heavy quark pair into dilepton [64]. ForDM or heavy quark systems, the Sommerfeld enhancementat finite temperature has been discussed in the literature,where the chemical equilibration rate is (i) estimated fromlinear response theory [37,65,66] and (ii) based on classicalrate arguments [67], is then inserted into the nonlinearBoltzmann equation for the DM number density “by hand.”Relying on those estimates, it has been shown that theoverclosure bound of the DM mass can be strongly affectedby the melting of bound states in a plasma environment[55–57]. However, strictly speaking, the linear responsetheory method is only applicable if the system is close tochemical equilibrium. Indeed the computation has been donein the imaginary-time formalism so far, capturing the physicsof thermal equilibrium. One of our goals is to obtain a moregeneral result directly from the real-time formalism, valid aswell for systems way out of equilibrium.Most of the necessary basics of the real-time method we

use are provided in Sec. II as a short review of out-of-equilibrium quantum field theory. Within this mathematicalframework, an exact expression for the DM number densityequation of our system is derived in Sec. III, where theSommerfeld-enhanced annihilation or decay rate at finitetemperature can be computed from a certain component of afour-point correlation function. We derive the equation of

BINDER, COVI, and MUKAIDA PHYS. REV. D 98, 115023 (2018)

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motion for the four-point correlation function on the real-timecontour in Sec. IV which becomes in its truncated form aBethe-Salpeter type of equation. Since we close the correla-tion functions hierarchy by truncation, the system of coupledequationswe have to solve contains only termswithDM two-and four-point functions. In Sec. V, we derive a simplesemianalytical solution of the four-point correlator undercertain assumptions valid for WIMP-like freeze-outs. Ourresult does not rely on linear response theory and it is thereforequite general applying also in case of large deviations fromchemical equilibrium. The limit of vanishing finite temper-ature corrections is taken in Sec. VI, showing the consistencybetween our general results and the classical Boltzmannequation treatment. Here, we also compare to the linearresponse theorymethodand clarify the assumptions needed toreproduce those results. Our main numerical results for thefinite temperature case are given in Sec. VII, both for a gaugetheory and for a Yukawa potential, and discussed in detail inSec. VIII. Finally, we conclude in Sec. IX.

II. REAL-TIME FORMALISM PREREQUISITES

The generalization of quantum field theory on theclosed-time-path (CTP) contour, or real-time formalism,is a mathematical method which allows one to describe thedynamics of quantum systems out of equilibrium.Prominent applications are systems on curved space-timeand/or systems having a finite temperature. In this work, weassume that the equilibration of DM in the early Universe isa fast process, and consequently, the initial memory effectsbefore the freeze-out process can be ignored. This leads tothe fact that the adiabatic assumption for such a system isan excellent approximation, motivating us to take theKeldysh-Schwinger prescription1 of the CTP contour, asillustrated in Fig. 1. The time contour C in the Keldysh-Schwinger prescription consists of two branches denotedby τþ and τ−. The upper time contour τþ ranges from the

initial time ti ¼ −∞ to t ¼ ∞while the lower contour τ− isconsidered to go from ∞ back to −∞. Therefore, times onthe τ− branch are said to be always later compared to thetimes on τþ. The time ordering of operator products on Ccan generically be written as

TC½O1ðt1Þ…OnðtnÞ�≡X

P

ð−1ÞFðpÞθCðtpð1Þ; tpð2ÞÞ…θCðtpðn−1Þ; tpðnÞÞ

×Opð1Þðtpð1ÞÞ…OpðnÞðtpðnÞÞ; ð2:1Þwhere the sum is over all the permutations P of the set ofoperators Oi and FðpÞ is the number of permutations offermionic operators. The unit step function and the deltadistribution on the Keldysh-Schwinger contour is defined as

θCðt1; t2Þ≡

8>>><>>>:

θðt1 − t2Þ if t1; t2 ∈ τþ

θðt2 − t1Þ if t1; t2 ∈ τ−

1 if t1 ∈ τ−; t2 ∈ τþ

0 if t1 ∈ τþ; t2 ∈ τ−

¼�θðt1 − t2Þ 0

1 θðt2 − t1Þ

�;

δCðt1; t2Þ ¼�δðt1 − t2Þ 0

0 −δðt1 − t2Þ

�: ð2:2Þ

Correlation functions, i.e., contour C-ordered operator prod-ucts averaged over all states where the weight is the densitymatrix of the system denoted by ρ, are defined by

hTCOðx1; x2;…; xnÞi≡ Tr½ρTCOðx1; x2;…; xnÞ�: ð2:3ÞLet us introduce commonly used notations and properties oftwo-point correlation functions of fermionic or bosonicoperator pairs relevant for this work. Because of the two-time structure, there are four possibilities to align the times x0

and y0 on C and hence four different components of a generaltwo-point function denoted by Gðx; yÞ, where in matrixnotation it can be written as

Gðx; yÞ≡ hTCψðxÞψ†ðyÞi¼ θCðx0; y0ÞhψðxÞψ†ðyÞi ∓ θCðy0; x0Þhψ†ðyÞψðxÞi

¼�Gþþðx; yÞ Gþ−ðx; yÞG−þðx; yÞ G−−ðx; yÞ

�: ð2:4Þ

FIG. 1. Keldysh-Schwinger approximation of the closed-time-path contour C, consisting of two time branches τþ and τ−.

1In ordinary QFT the initial vacuum state Ω appearing incorrelation functions hΩjinT½Oðx;…Þ�jΩiin is equivalent up to aphase to the final vacuum state. For this special situation theoperators are ordered along the “flat” time axis ranging fromtin ¼ −∞ to tout ¼ ∞. By means of Lehmann-Symanzik-Zim-mermann (LSZ) reduction formula it is then possible to relatecorrelation functions to the S matrix and compute cross sections.This in-out formalism breaks down once, e.g., the initial vacuumis not equivalent to the final state vacuum. An expandingbackground or external sources can introduce such a timedependence. In our work, there are mainly two sources ofbreaking the time translation invariance. First, since we have athermal population, we consider traces of time-ordered operatorproducts, where the trace is taken over all possible states. Themany particle states are in general time dependent. Second, wehave a density matrix next to the time ordering. The CTP, or, in-informalism we adopt in this work can be, pragmatically speaking,seen as just a mathematical way of how to deal with such moregeneral expectation values. The Keldysh description of the CTPcontour applies if initial correlations can be neglected and werefer for a more detailed discussion and limitation to [68].

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Here, Gσxσy means x0 ∈ τσx and y0 ∈ τσy with σi ¼ � fori ¼ x, y and the four different components of Gðx; yÞ aredefined as

G−þðx; yÞ≡ hψðxÞψ†ðyÞi; ð2:5Þ

Gþ−ðx; yÞ≡ ∓ hψ†ðyÞψðxÞi; ð2:6Þ

Gþþðx; yÞ≡ θðx0 − y0ÞG−þðx; yÞ þ θðy0 − x0ÞGþ−ðx; yÞ;ð2:7Þ

G−−ðx; yÞ≡ θðx0 − y0ÞGþ−ðx; yÞ þ θðy0 − x0ÞG−þðx; yÞ;ð2:8Þ

where −ðþÞ on the r.h.s. of the second line applies forfermionic (bosonic) field operators. From these definitionsone can recognize that not all components are independent,namely the following relation holds:

Gþþðx; yÞ þ G−−ðx; yÞ ¼ Gþ−ðx; yÞ þ G−þðx; yÞ: ð2:9Þ

Furthermore, let us introduce retarded and advanced corre-lators defined by

GRðx; yÞ≡ θðx0 − y0Þ½G−þðx; yÞ − Gþ−ðx; yÞ�¼ Gþþðx; yÞ −Gþ−ðx; yÞ¼ −G−−ðx; yÞ þ G−þðx; yÞ; ð2:10Þ

GAðx; yÞ≡ −θðy0 − x0Þ½G−þðx; yÞ −Gþ−ðx; yÞ�¼ Gþþðx; yÞ − G−þðx; yÞ¼ −G−−ðx; yÞ þ Gþ−ðx; yÞ; ð2:11Þ

as well as the spectral function given by

Gρðx; yÞ≡ GRðx; yÞ − GAðx; yÞ ¼ G−þðx; yÞ − Gþ−ðx; yÞ:ð2:12Þ

From these definitionswe canderive further useful properties:

GAðx; yÞ ¼ −½GRðy; xÞ�†;Gþ−ðx; yÞ ¼ ½Gþ−ðy; xÞ�†;G−þðx; yÞ ¼ ½G−þðy; xÞ�†: ð2:13Þ

In the case of free (unperturbed) propagatorsG0, the followingsemigroup property holds:

GR0 ðx; yÞ ¼

Zd3zGR

0 ðx; zÞGR0 ðz; yÞ; for tx > tz > ty:

ð2:14ÞThis equality can be verified by noticing that for those timeconfigurations the correlators are proportional to on-shellplane waves in Fourier space. Note that there is no timeintegration in the above equation. Together with the relationsin Eq. (2.13) further semigroup properties can be derived andall important ones are summarized for the use in subsequentsections in Appendix A.As an example, the time integration over the Schwinger-

Keldysh contourC of products of correlators can bewritten as

Cþþðx; yÞ ¼�Z

z0∈Cdz0

Zd3zAðx; zÞBðz; yÞ

�x0;y0∈τþ

¼��Z

τþdz0 þ

Zτ−dz0

�Zd3zAðx; zÞBðz; yÞ

�x0;y0∈τþ

¼Z

∞

−∞dz0

Zd3zAþþðx; zÞBþþðz; yÞ þ

Z−∞

∞dz0

Zd3zAþ−ðx; zÞB−þðz; yÞ

¼Z

∞

−∞d4zðAþþðx; zÞBþþðz; yÞ − Aþ−ðx; zÞB−þðz; yÞÞ: ð2:15Þ

Equation (2.15) is called a Lagereth rule and it is straightforward to work out similar rules for, e.g., different components ordouble integrations of higher-order products of Keldysh-Schwinger correlators as they will appear later in this work. Let usmove on and define Wigner coordinates according to

T ¼ ðx0 þ y0Þ=2; t ¼ x0 − y0; ð2:16ÞR ¼ ðxþ yÞ=2; r ¼ x − y: ð2:17Þ

In the second line all variables are 3-vectors. The Wigner-transformed correlators are defined as

Gðt; r;R; TÞ≡GðT þ t=2;Rþ r=2; T − t=2;R − r=2Þ¼ Gðx0x; y0yÞ: ð2:18Þ


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In all computations, the tilde will be dropped such that we can write for the Fourier transformation of Gðx; yÞ:

Gðω;p;R; TÞ ¼Z

dtd3reiðωt−p·rÞGðt; r;R; TÞ: ð2:19Þ

One of the great advantages of separating microscopic (t, r) and macroscopic (T, R) variables according to the Wignertransformation is that Fourier transformations of integral expressions can be considerably simplified by using the gradientexpansion. For example, Eq. (2.15) in Fourier space can be written as

Cþþðω;p;R; TÞ ¼ Aþþðω;p;R; TÞGABBþþðω;p;R; TÞ − Aþ−ðω;p;R; TÞGABB−þðω;p;R; TÞ≃ Aþþðω;p;R; TÞBþþðω;p;R; TÞ − Aþ−ðω;p;R; TÞB−þðω;p;R; TÞ; ð2:20Þ

GAB ≡ e−ið∂AT∂Bω−∂Aω∂BT−∇AR·∇B

pþ∇Ap ·∇B

RÞ=2 ≃ 1: ð2:21Þ

Throughout this work, such exponentials containing deriv-atives are alwaysapproximatedas in the last line, defining theleading order term of the gradient expansion. For homo-geneous and isotropic systems the correlators do not dependon R and thus for the spatial part the leading order term isexact. For a discussion of the validity of the leading orderterm of the temporal part we refer to [69]. Let us emphasizehere, that for typical DM scenarios the leading order term isalways assumed to be a very good approximation.Next, important properties of two-point correlators in

thermal equilibrium are provided. Under this assumption,different components of correlators become related whichare otherwise independent. For a system being in equilib-rium (here we consider kinetic as well as chemicalequilibrium), the density matrix takes the form

ρ ∝ e−βH; ð2:22Þ

whereH is the Hamiltonian of the system and β factor is theinverse temperature T of the system. The density matrix inthermal equilibrium can be formally seen as a time evolutionoperator, where the inverse temperature is regarded as anevolution in the imaginary time direction. Making use of thecyclic property of the trace it can be shown that under theassumption of equilibrium the components are related via

G−þðx0 − y0Þ ¼∓ Gþ−ðx0 − y0 þ iβÞ: ð2:23Þ

This important property is called the Kubo-Martin-Schwinger (KMS) relation, where − (þ) applies for two-point correlators of fermionic (bosonic) operators.Furthermore, in equilibrium the correlators should dependonly on the difference of the time variables due to timetranslation invariance. Consequently, the KMS condition inFourier space reads

G−þðω;pÞ ¼ ∓eβωGþ−ðω;pÞ: ð2:24Þ

From this condition, various equilibrium relations follow:

Gþ−ðω;pÞ ¼∓nF=BðωÞGρðω;pÞ;G−þðω;pÞ ¼ ½1 ∓ nF=BðωÞ�Gρðω;pÞ; ð2:25Þ

Gþþðω; pÞ ¼ GRðω;pÞ þGAðω;pÞ2

þ�1

2∓ nF=BðωÞ

�Gρðω;pÞ; ð2:26Þ

where the Fermi-Dirac or Bose-Einstein phase-space den-sities are given by nF=BðωÞ ¼ 1=ðeβω � 1Þ. Thus in equilib-rium, all correlator components can be calculated from theretarded/advanced components, where the spectral functionGρ is related to those via Eq. (2.12).General out-of-equilibrium observables, like the dynamic

of the number density or spectral information of the system,can be directly inferred from the equation of motions (EoM)of the corresponding correlators. Throughout this work, wederive the correlator EoM from the invariance principle ofthe path integral measure under infinitesimal perturbations ofthe fields. The equivalence of CTP correlators and the pathintegral formulation is given by

hTCOðx1;x2;…;xnÞi¼Z

dμiρðμiÞZμi

½dμ�Oðx1;x2;…;xnÞ

×eiRx∈C

LðxÞ ð2:27Þ

and the action on the CPT contour is S ¼ Rx∈C LðxÞ.

μ collectively represents the fields, and ρ stands for a statethat could be either pure or mixed, as in Eq. (2.3). Thesecond integral in Eq. (2.27) is a path integral with aboundary condition of μi at the initial time ti that we taketo−∞ in the Schwinger-Keldysh prescription of the contour,and the first one takes the average of μi with the weight ofρðμiÞ. Now, to derive the EoM for two-point correlators, letus consider an infinitesimal perturbation O0† ¼ O† þ ϵ,satisfying ϵðtiÞ ¼ 0. By relying on the measure-invarianceprinciple under this transformation, one obtains the EoM ofthe two-point correlators from


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hO0†ðyÞi¼hO†ðyÞiþϵðxÞþiZx∈C

ϵðxÞ�TC

δSδLO†ðxÞO

†ðyÞ�

þOðϵ2Þ ð2:28Þ

⇒ 0 ¼ δCðx; yÞ þ i

�TC

δSδLO†ðxÞO

†ðyÞ�; ð2:29Þ

where δL represents a derivative acting from the left. Thesame procedure can be applied to the case ofO, as well as forderiving the EoM of higher correlation functions. Therelation between the abstract EoM of correlators and differ-ential equations for observables will be part of the nextsection. In general, a correlator EoM depends on higher andlower correlators which is called the Martin-Schwingerhierarchy. For systems where the coupling expansion isappropriate, it might be sufficient to work in the one-particleself-energy approximation, where the EoM are closed interms of two-point functions, and the kinetic equations cansystematically be obtained by expanding the DM self-energyperturbatively in the coupling constant. The kinetic equa-tions of two-point functions in the self-energy approximationare also known as the Kadanoff-Baym equations. Forexample, at next-to-leading order (NLO) in the self-energyexpansion of the two-point correlators the standardBoltzmann equation is recovered.Finite temperature corrections to nonperturbative systems,

e.g., Sommerfeld-enhancedDMannihilations or bound-statedecays, where a subclass of higher-order diagrams becomescomparable to the leading order (LO) in vacuum, are lessunderstood in the CTP formalism. The strategy in the nextsectionwill be to address this issue by going beyond the one-particle self-energy approximation [69]. More precisely, wederive the exact Martin-Schwinger hierarchy of our particlesetup in the CTP formalism by using Eq. (2.29) and truncatethe hierarchy at the six-point function level. The system ofequations is then closed with respect to two- and four-pointfunctions. This approach allows us to account for theresummation of the Coulomb diagrams and their finitetemperature corrections at the same time. Furthermore, weshow how to extract the DM number density equation fromthe EoM of two-point functions and that it depends on thefour-point correlator. The complication is that the differentialequation of the four-point correlator is coupled to the two-point correlator and in subsequent sections we solve thiscoupled systems of equations.

III. EQUATIONS OF MOTION INREAL-TIME FORMALISM FOR

NONRELATIVISTIC PARTICLES IN ARELATIVISTIC PLASMA ENVIRONMENT

Throughout this paper, we consider the following min-imal scenario capturing important effects to study long-range force enhanced DM annihilations and bound statesunder the influence of a hot and dense plasma environment:

L ¼ χði=∂ −MÞχ þ gχ χγμχAμ þ ψði=∂ −mÞψ þ gψ ψγμψAμ

−1

4FμνFμν: ð3:1Þ

The particles of the equilibrated plasma environment withtemperature T are the Abelian mediators Aμ and the lightfermionic particles ψ with mass m ≪ T. Fermionic DM χis assumed to be nonrelativistic, i.e., M ≫ T. All fer-mionic particles are considered to be of Dirac type. Weassume the mediator to be massless; however, we providethe final results also for the case of a massive Aμ withmass mV ≪ M.Let us illustrate how we can get the DM nonrelativistic

effective action in the thermal medium of light particles. Itis obtained in two steps. First, hard modes of p≳M areintegrated out. In this limit, the DM four-component spinorχ splits into two parts, a term for the particle denoted by thetwo-component spinor η and a term for the antiparticledenoted by ξ. Second, we assume that DM does notinfluence the plasma environment during the freeze-outprocess. This is typically the case since the DM energydensities are smaller than that of light particles at thisepoch. And thus, we may also integrate out the plasmafields by assuming they remain in thermal equilibrium. Theresulting effective action on the CTP contour for particle ηand anti-particle ξ DM is given by

SNR½η;ξ� ¼Zx∈C

η†ðxÞ�i∂tþ

∇2

2M

�ηðxÞþξ†ðxÞ

�i∂t−

∇2

2M

�ξðxÞ

þZx;y∈C

ig2χ2JðxÞDðx;yÞJðyÞ

þ iO†sðxÞΓsðx;yÞOsðyÞ: ð3:2Þ

Dark matter long-range force interactions are encoded inthe first term of the interactions in Eq. (3.2). This termincludes the current and the full two-point correlator of theelectric potential on the CTP contour which are defined by

JðxÞ≡ η†ðxÞηðxÞ þ ξ†ðxÞξðxÞ;Dðx; yÞ≡ hTCA0ðxÞA0ðyÞi; ð3:3Þ

respectively. The last term in Eq. (3.2) describes theannihilation part and we only keep the s-wave contribution

OsðxÞ≡ξ†ðxÞηðxÞ;

Γsðx;yÞ≡πðα2χ þαχαψÞM2

�δ4ðx−yÞ 0

2δ4ðx−yÞ δ4ðx−yÞ

�; ð3:4Þ

with the fine-structure constant being αi ≡ g2i =4π, andsummation over the spin indices are implicit. Γs is shownin the matrix representation of the CTP formalism; see e.g.,Eq. (2.4) in previous section. Hence the delta functions onthe right-hand side are defined on the usual real-time axis.


115023-6

Similar to the vacuum theory, the annihilation part Γs canbe computed by cutting the box diagram (containing twoAμ) and the vacuum polarization diagram (containing oneAμ and a loop of light fermions ψ), where now allpropagators are defined on the CPT contour. Finite temper-ature corrections to these hard processes in Γs areneglected2 and for a derivation we refer to Appendix B.In our effective action Eq. (3.2), we have discarded

higher order terms in ∇=M (like magnetic interactions) andalso interactions with ultrasoft gauge bosons,3 since wefocus on threshold singularities of annihilations at theleading order in the coupling gχ and the velocity ∇=M.Furthermore, our effective field theory is non-Hermitianbecause we have integrated out (or traced out) hard andthermal degrees of freedom (d.o.f.). The first source of non-Hermite nature is the annihilation term which originatesfrom the integration of hard d.o.f. A similar term wouldalso be present in vacuum [10,11,46] and belongs to the ++component of Γ. Thus, as a first result we have generalizedthe annihilation term towards the CTP contour. Anotherone stems from the gauge boson propagatorD that encodesinteractions with the thermal plasma. While the annihila-tion term containing Γs in our action breaks the numberconservation of DM, the interaction term containing D cannot. From this observation, one may anticipate that the non-Hermitian potential contributions of the gauge bosonpropagator never lead to a violation of the DM particleor antiparticle number conservation. Later, we will showthis property directly from the EoM, respecting the globalsymmetries of our action.In the next sections we proceed as in the following. First,

we compute the finite temperature one-loop correctionscontained in the potential term D explicitly. Since the

number density of DM becomes Boltzmann suppressed inthe nonrelativistic regime of the freeze-out process, thedominant thermal loop contributions arise from the rela-tivistic species ψ . This implies that we can solve for Dindependently of the DM system since we assumeDM does not modify the property of the plasma. Thecorrection terms for the DM self-interactions are screeningeffects on the electric potential, as well as imaginarycontributions arising from soft DM-ψ scatterings, derivedand discussed in detail in Sec. III A. Second, in Sec. III Bthe kinetic equations for the DM correlators are derived. Weshow how to extract from these equations the numberdensity equation, including finite temperature correctedprocesses for the negative energy spectrum (bound-statedecays) as well as for the positive energy contribution(Sommerfeld-enhanced annihilation) in one singleequation.

A. Thermal corrections to potential term

In this section, we briefly summarize how the electriccomponent of the mediator propagator D gets modifiedby the thermal presence of ultrarelativistic ψ fields. Theplasma environment is regarded to be perturbative and intheone-particle self-energy frameworkwe canwrite down theDyson equation on the Schwinger-Keldysh contour for themediator:

Dμνðx;yÞ¼D0μνðx;yÞ− i

Zw;z∈C

D0μαðx;wÞΠαβðw;zÞDβνðz;yÞ;

ð3:5ÞΠμνðx; yÞ ¼ ð−iÞg2ψð−1ÞTr½γμSðx; yÞγνSðy; xÞ�; ð3:6Þ

where Sðx; yÞ≡ hTCψðxÞψðyÞi0 are unperturbed ψcorrelators. Aμ and ψ are assumed to be in equilibrium andthus, according to the discussion in Sec. II, we only need tocompute the retarded/advanced propagators. From those, wecan construct all other components by using the KMScondition. The Dyson equation for retarded (advanced)mediator-correlator can be obtained by subtracting the þ −ð−þÞ component of Eq. (3.5) from theþþ component of thesame equation. In momentum space this results in

DR=Aμν ¼ ðDR=A

μν Þ0 − iðDR=Aμα Þ0Παβ

R=ADR=Aβν ; ð3:7Þ

where the mediator’s retarded self-energy is defined as

ΠμνR ðPÞ ¼ Πμν

þþ − Πμνþ−

¼ ig2ψ

Zd4Kð2πÞ4 ðTr½γ

μSþþðK − PÞγνSþþðKÞ�

− Tr½γμSþ−ðK − PÞγνS−þðKÞ�Þ: ð3:8ÞA sketch of efficiently calculating the thermal one-loopEq. (3.8) is provided in Appendix C. In the computationwe utilize thehard thermal loop (HTL) approximation [71] to

2Assuming free plasma field correlators in the computation ofΓs is a good approximation since the energy scale of the hardprocess is ∼M which is much larger for nonrelativistic particlesthan typical finite temperature corrections being of the order∼gψT. Consequently, the dominant thermal corrections should bein the modification of the long-range force correlator D, wherethe typical DM momentum-exchange scale enters which is muchlower compared to the annihilation scale.

3To fully study the out-of-equilibrium dynamics of the bound-state formation [21,22,42,70] at late times of the freeze-outprocess, it is necessary to include emission and absorption viaultrasoft gauge bosons, e.g., via an electric dipole operator. Wedrop for simplicity ultrasoft contributions and discuss in detail thelimitation of our approach later in this work; see Sec. VI. Notehere that at high enough temperature those processes are typicallyefficient, leading just to the ionization equilibrium amongbounded and scattering states. As long as ionization equilibriumis maintained, the effective action we use is sufficient enough todescribe Sommerfeld enhanced annihilation and bound-statedecay at finite temperature. To estimate the time when theionization equilibrium is violated concretely, we have to takeinto account these processes in the thermal plasma, which will bepresented elsewhere. In vacuum, ionization equilibrium starts tobecome violated when the decay width of the lowest bound stateexceeds the ionization rate.


115023-7

extract leading thermal corrections.4 In the HTL approxi-mation we are allowed to resum the self-energy contri-butions of the retarded/advanced component and the resultis gauge independent. We work in the noncovariantCoulomb gauge which is known to be fine at finitetemperature since Lorentz invariance is anyway brokenby the plasma temperature. We find for the dressedlongitudinal component μ ¼ ν ¼ 0 of the mediator propa-gator in the HTL resummed self-energy approximation andin the Coulomb gauge the following:

DR;Aðω;pÞ ¼ −iω2 − p2 þ Π00

R;Aðω;pÞ � isignðωÞϵ ; ð3:9Þ

Π00R;Aðω;pÞ ¼ −m2

D

�1 −

ω

2jpj ln��ωþ jpj

ω − jpj��

� iω

jpjπ

2θðjpj2 − ω2Þ

�; ð3:10Þ

where we introduced the Debye screening massm2

D ¼ g2ψT2=3. One can recognize that there is correctionto the real part of the mediator propagator as well as abranch cut for spacelike exchange. Using the equilibriumrelation Eq. (2.26), the þþ component of D in the staticlimit reads

limω→0

Dþþðω;pÞ ¼ limω→0

�DRðω;pÞ þDAðω;pÞ

2

þ�1

2þ nBðωÞ

�Dρðω;pÞ

�

¼ ip2 þm2

Dþ π

Tjpj

m2D

ðp2 þm2DÞ2

; ð3:11Þ

while for a massive mediator we have simply

limω→0

Dþþðω;pÞ¼ ip2þm2

V þm2Dþπ

Tjpj

m2D

ðp2þm2V þm2

DÞ2:

ð3:12Þ

The static Dþþ component is of special importance fordescribing DM long-range interactions in a plasma envi-ronment as we will see later in this work. The first term inEqs. (3.11) and (3.12) will result in a screened Yukawapotential after Fourier transformation while the secondterms will lead to purely imaginary contributions.Physically, the latter part originates from the scatteringof the photon with plasma fermions, leading to a dampingrate [72]. Indeed in the quasiparticle picture, the mediatorhas a limited propagation time within the plasma, whichlimits as well the coherence of the mediator exchangeprocesses. For what regards the DM particles, this termwill later give rise to DM-ψ scattering with zero energytransfer, leading also to a thermal width for the DM states.In the following sections, we try to keep generality andwork in most of the computations with the unspecifiedform Dðx; yÞ and take just at the very end the static andHTL limit. Let us finally remark that the simple form ofEq. (3.12) allows us to achieve semianalytical results forthe DM annihilation or decay rates in the presence of athermal environment.

B. Exact DM number density equation fromcorrelator equation of motion

The main purpose of this section is to derive the equationfor the DM number density directly from the exact EoM ofour nonrelativistic action. Defining the DM particle andantiparticle correlators as

Gηðx; yÞ≡ hTCηðxÞη†ðyÞi; ð3:13Þ

Gξðx; yÞ≡ hTCξðxÞξ†ðyÞi; ð3:14Þ

we derive respective EoM from the path-integral formal-ism, as briefly explained at the end of Sec. II, for thenonrelativistic effective action SNR given in Eq. (3.2):�i∂x0 þ

∇2x

2M

�Gηðx; yÞ

¼ iδCðx; yÞ − ig2χ

Zz∈C

Dðx; zÞhTCηðxÞJðzÞη†ðyÞi

− iZz∈C

Γsðx; zÞhTCξðxÞfξ†ðzÞηðzÞgη†ðyÞi; ð3:15Þ

�−i∂y0 þ

∇2y

2M

�Gηðx; yÞ


Zz∈C

Dðy; zÞhTCηðxÞJðzÞη†ðyÞi

− iZz∈C

Γsðz; yÞhTCηðxÞfη†ðzÞξðzÞgξ†ðyÞi; ð3:16Þ

4Let us briefly summarize here the assumptions of the HTLapproximation. First, we drop all vacuum contributions and onlykeep temperature dependent parts. Second, we assume the externalenergy P0 and momentum p to be much smaller than typical loopmomentum k which is of the order temperature (see Appendix C).The discussion of the validity of the HTL approximation dependson where the dressed mediator correlator is attached to. One cannot naively argue for the case where one would attach it to the DMcorrelator that the external momentum p is the DM momentumwhich is of course much larger than temperature, thus invalidatingthe HTL approximation by this argumentation. For example, in ourcase the dressed mediator correlator enters the DM single-particleself-energy (see Fig. 2 and Sec. V C) and the dominant energy andmomentum region in the loop diagram is where HTL effectivetheory is valid.


115023-8

�i∂x0 −

∇2x

2M

�Gξðx; yÞ


Zz∈C

Dðx; zÞhTCξðxÞJðzÞξ†ðyÞi

− iZz∈C

Γsðx; zÞhTCηðxÞfη†ðzÞξðzÞgξ†ðyÞi; ð3:17Þ�−i∂y0 −

∇2y

2M

�Gξðx; yÞ


Zz∈C

Dðy; zÞhTCξðxÞJðzÞξ†ðyÞi

− iZz∈C

Γsðy; zÞhTCξðxÞfξ†ðzÞηðzÞgη†ðyÞi: ð3:18Þ

The anticipated structure in Eqs. (3.15)–(3.18) shows thedependence of the two-point correlators on higher corre-lation functions. Here, the curly brackets stand for thesummation over the spin indices and J is the current asalready introduced in Eq. (3.3). It might be helpful tomention that we used a special property of two-pointfunctions of Hermitian bosonic field operators: Dðx; yÞ ¼

Dðy; xÞ. This exact property can be verified directly fromthe definition in Eq. (3.3).In the following, the number density equation of particle

and antiparticle DM is derived from this set of differentialequations. First of all, we would like to clarify what is thenumber density in terms of fields appearing in SNR inEq. (3.2). For this purpose, let us switch off the annihilationterm Γs → 0 in SNR and seek for conserved quantities. Inthis limit, the theory has the following global symmetries:η ↦ eiθηη and ξ ↦ e−iθξξ. The associated Noether currentsfor the DM particle and antiparticle are

JμηðxÞ ¼�η†ðxÞηðxÞ; 1

2iMη†ðxÞ∇↔ηðxÞ

�;

JμξðxÞ ¼�ξðxÞξ†ðxÞ; 1

2iMξðxÞ∇↔ξ†ðxÞ

�: ð3:19Þ

The thermal-averaged zeroth component is the numberdensity and the average over spatial component results inthe current density. We obtain the differential equation forthe two DM currents directly from the two-point functionEoM, by subtracting Eqs. (3.16) from Eqs. (3.15) andEqs. (3.18) from Eqs. (3.17), and by taking the spin-traceand the limit y → x. For the particle DM, we obtain as anintermediate result after all these steps:

i∂μhJμηi ¼ −�i∂x0 þ i∂y0 þ

∇2x

2M−

∇2y

2M

�TrGηðx; yÞjy→x

¼ þig2χ

Zz∈C

Dðx; zÞ½hTCfηðxÞη†ðxÞgJðzÞi − hTCfηðxÞη†ðxÞgJðzÞi�

− iZz∈C

½Γsðx; zÞhTCfηðzÞξ†ðzÞgfξðxÞη†ðyÞgi − Γsðz; yÞhTCfηðxÞξ†ðyÞgfξðzÞη†ðzÞgi�y→x: ð3:20Þ

The trace and the curly brackets indicate the summationover the spin indices. It is important to note that the first linein the second equality cancels out, even in the case of afully interacting correlator D including finite temperaturecorrections. Thus, we confirm from the EoM that, e.g., non-Hermitian potential corrections arising from soft thermalDM-ψ scatterings in the HTL approximation of D [seeEq. (3.11)], never violate the current conservation of eachindividual DM species. For a homogeneous and isotropicsystem (vanishing divergence of current density) this wouldmean that the individual number densities of particles andantiparticles do not change by self-scattering processes, realphysical DM-ψ scatterings, soft DM-ψ scatterings or otherfinite temperature corrections leading to potential contri-butions in D.It can be recognized, that the current conservation is only

violated by the annihilation term Γs in the last line inEq. (3.20), since this contribution does not cancel to zero.This term can be simplified by using Eq. (3.4) and by fixingthe time component x0 to either τþ or τ−. We have explicitly

checked that both choices of x0 lead to the same final result.With the definition of the four-point correlator on theclosed-time-path contour

Gηξ;sðx; y; z; wÞ≡ hTCηiðxÞξ†i ðyÞξjðwÞη†jðzÞi; ð3:21Þ

we obtain our final form of the current equations:

∂μhJμηðxÞi ¼ −2πðα2χ þ αχαψÞ

M2Gþþ−−

ηξ;s ðx; x; x; xÞ; ð3:22Þ

∂μhJμξðxÞi ¼ −2πðα2χ þ αχαψÞ

M2Gþþ−−

ηξ;s ðx; x; x; xÞ: ð3:23Þ

The current conservation is only violated by contributionscoming from Γs. This is consistent with the expectationsfrom the symmetry properties of the actionwhen annihilationis turned on. Namely, only a linear combination of bothglobal transformations leaves the action invariant whichleads to the conservation of ∂μðJμη − JμξÞ ¼ 0, which is


115023-9

nothing but the DM asymmetry current conservation. Theconservation of the totalDMnumber density, ∂μðJμη þ JμξÞ, isviolated by the annihilation term.Before we discuss the four-point correlator appearing in

Eqs. (3.22) and (3.23) in detail, let us now assume aFriedman-Robertson-Walker universe and make the con-nection to the Boltzmann equation for the number densitythat is typically adopted in the literature when calculatingthe thermal history of the dark matter particles. First, thespatial divergence on the left-hand side of the currentEqs. (3.22) and (3.23) vanishes due to homogeneity andisotropy. Second, the adiabatic expansion of the back-ground introduces a Hubble expansion termH. Third, it canbe seen from the sign of the right-hand side of the currentequations that only a DM loss term occurs. The productionterm is missing because we have assumed a priori, whenderiving the nonrelativistic action, that the DM mass ismuch larger than the thermal plasma temperature. Withinthis mass-to-infinity limit the DM production term is set tozero in the computation of the annihilation term Γs and notexpected to occur. Let us therefore add on the r.h.s. of thecurrent equations a posteriori the production term ofthe DM via the assumption of detailed balance, resultingin the more familiar number density equations:

_nη þ 3Hnη

¼ −2ðσvrelÞ½Gþþ−−ηξ;s ðx; x; x; xÞ − Gþþ−−

ηξ;s ðx; x; x; xÞjeq�;ð3:24Þ

_nξ þ 3Hnξ

¼ −2ðσvrelÞ½Gþþ−−ηξ;s ðx; x; x; xÞ − Gþþ−−

ηξ;s ðx; x; x; xÞjeq�:ð3:25Þ

The tree-level s-wave annihilation cross section of oursystemwas defined as ðσvrelÞ ¼ πðα2χ þ αχαψÞ=M2 and jeq inthe last term means the evaluation at thermal equilibrium.Note that in the CTP formalism a cross section strictlyspeaking does not exist. The reason why this result is equalto the vacuum computation is because we computed theannihilation part Γs at the leading order, where it is expectedthat zero and finite temperature results should coincide. Thecorrelation function Gþþ−−

ηξ;s however is fully interacting. Wesummarizewith two concluding remarks on our main results:

(i) Sommerfeld-enhancement factorat finite temperature:One of our findings is that the Sommerfeld factoris contained in a certain component of the interactingfour-point correlation function, namely Gþþ−−

ηξ;s . Thisresult is valid for a generic out-of-equilibrium state ofthe dark matter system. The remaining task is to find asolution for this four-point correlator. As we will seelater, the solution can be obtained from the Bethe-Salpeter equation on the CTP contour, derived inthe next section. For example, expanding theBethe-Salpeter equation to zeroth order in the DM

self-interactions 2Gþþ−−ηξ;s ðx;x;x;xÞ≃−2Gþ−

η ðx;xÞ×Gþ−

ξ ðx;xÞ¼nηnξ and inserting this into Eqs. (3.24)and (3.25)would result in awell-known expression forthe number density equation of the DM particles withvelocity-independent annihilation.Aswewill see later,higher terms in the interaction or a fully nonperturba-tive solution contain the finite temperature correctednegative and positive energy spectrum. In otherwords,Gþþ−−

ηξ;s contains both, the bound-state and scatteringstate contributions at the same time and at finitetemperature they turn out to be not separable as it issometimes done invacuum computations. Bound statecontributions will automatically change the crosssection into a decay width and thus, nη appearing onthe l.h.s. of Eq. (3.24) is the total number of η particlesincluding the ones in the bound states and similarinterpretation for the antiparticle ξ.

(ii) Particle number conservation: In Sec. III A, we haveseen that the thermal corrections to the mediatorpropagator D can contain, next to the real Debyemass, an imaginary contribution. It was shown thatthese non-Hermitian corrections to the potential neverviolate the particle number conservation due to theexact cancellation of the second line in Eq. (3.20).This was expected from the beginning, since, whenswitching off the annihilation Γs → 0, the nonrela-tivistic action in Eq. (3.2) has two global symmetriesη ↦ eiθηη and ξ ↦ e−iθξξ. The conserved quantitiesare the particle and antiparticle currents in Eqs. (3.22)and (3.23) in the limit Γs → 0 (vanishing r.h.s.).When annihilation is included, the nonrelativisticaction is only invariant if both global transformationsare performed at the same time, resulting in theconserved asymmetry current Jη − Jξ. We concludethat thermal corrections can never violate thesesymmetries, even not at higher loop level. On theother hand, the solution of the Sommerfeld factor iscontained in Gþþ−−

ηξ;s and hence the annihilation ratewill depend on the thermal loop corrected long-rangemediator D, as we will see in the next section.

IV. TWO-TIME BETHE-SALPETER EQUATIONS

The exact number density Eq. (3.24) depends on theKeldysh four-point correlation function Gþþ−−

ηξ;s ðx; x; x; xÞ.In this section, we derive the system of closed equation ofmotion needed in order to obtain a solution for this four-point function, including the full resummation of Coulombdivergent ladder diagrams. The result will be a coupled setof two-time Bethe-Salpeter equations on the Keldyshcontour as given by the end of this section, Eqs. (4.20)–(4.21). They apply in general for out-of-equilibrium sit-uations and include in their nonperturbative form also thebound-state contributions if present. In order to arrive atthose equations, a set of approximations and assumptions is


115023-10

needed. We therefore would like to start from the beginningin deriving those equations, which might lead to a betterunderstanding of their limitation.In the first simplification, we treat the annihilation term

Γs as a perturbation and ignore it in the followingcomputations, since the leading order term in the annihi-lation part is already contained in Eq. (3.24). The exactset of EoMs for two- and four-point functions in the limitΓs → 0 are given by�i∂x0 þ

∇2x

2M

�Gηðx; yÞ ¼ iδCðx; yÞ − ig2χ

Zz∈C

Dðx; zÞ

× ½Gηξðx; z; y; zÞ −Gηηðx; z; y; zÞ�;ð4:1Þ�

i∂x0 þ∇2

x

2M

�Gξðx; yÞ ¼ iδCðx; yÞ − ig2χ

Zz∈C

Dðx; zÞ

× ½Gηξðz; x; z; yÞ −Gξξðx; z; y; zÞ�;ð4:2Þ�

i∂x0 þ∇2

x

2M

�Gηξðx; y; z; wÞ

¼ iδCðx; zÞGξðy; wÞ

− ig2χ

Zx∈C

Dðx; xÞhTCηðxÞJðxÞξ†ðyÞξðwÞη†ðzÞi; ð4:3Þ

where we will work for the rest of this work with theconjugate antiparticle correlator Gξ and, here, the spin-uncontracted correlators are defined as

Gξðx; yÞ≡ hTCξ†ðxÞξðyÞi; ð4:4Þ

Gηξðx; y; z; wÞ≡ hTCηðxÞξ†ðyÞξðwÞη†ðzÞi; ð4:5Þ

Gηηðx; y; z; wÞ≡ hTCηðxÞηðyÞη†ðwÞη†ðzÞi; ð4:6Þ

Gξξðx; y; z; wÞ≡ hTCξ†ðxÞξ†ðyÞξðwÞξðzÞi: ð4:7Þ

The EoMs for the two-point functions Eqs. (4.1) and (4.2)are equivalent to Eqs. (3.15) and (3.17) in the limit Γs → 0,and we have just rewritten them in terms of the four-pointcorrelators and conjugate antiparticle propagator, defined inEqs. (4.4)–(4.7). In our notation, the spinor indices of theoperators having equal space-time arguments in the four-point correlators are summed and J is the current as definedin Eq. (3.3). The different conventions for the ηη and ξξ

four-point correlators are because η† and ξ are the creationoperators. From the exact differential equation of the four-point correlator in Eq. (4.3), it can be seen that the correlatorhierarchy is still not closed yet, since it depends on the six-point function. We close this hierarchy of correlators bytruncating the six-point function at the leading order:�

i∂y0 þ∇2

y

2M

�hTCηðxÞJðxÞξ†ðyÞξðwÞη†ðzÞi

≃ iδCðy; wÞ½Gηξðx; x; z; xÞ −Gηηðx; x; z; xÞ�− iδCðx; yÞGηξðx; y; z; wÞ; ð4:8Þ

i.e., only the integral kernel of the six-point functioncontaining the eight-point function was dropped. The setof correlator differential equations is now closed under thistruncation procedure. A fully self-consistent solutionrequires in principle to solve the equations for the fivecorrelators Gη, Gξ, Gηξ; Gηη; Gξξ simultaneously. This isbeyond the scope of this work and we have to furtherapproximate the system in order to obtain at least a simplesemianalytical solution at the end.The solution of our target component Gηξ can formally

be decoupled from the solution of Gηη and Gξξ byapproximating the latter two quantities as the leading ordercontribution (dropping the integral kernel, known asHartree-Fock approximation). Then, one can recognize that

½Gηξðx; z; y; zÞ −Gηηðx; z; y; zÞ�≃ ½Gηðx; yÞGξðz; zÞ − Gηðx; yÞGηðz; zÞ þ Gηðx; zÞGηðz; yÞ�¼ Gηðx; zÞGηðz; yÞ; ð4:9Þ

½Gηξðz; x; z; yÞ −Gξξðx; z; y; zÞ�≃ ½Gξðx; yÞGηðz; zÞ − Gξðx; yÞGξðz; zÞ þ Gξðx; zÞGξðz; yÞ�¼ Gξðx; zÞGξðz; yÞ: ð4:10Þ

In both equations, the last step is a strict equality only if theDM particleGηðz; zÞ ¼ −nη and antiparticle Gξðz; zÞ ¼ −nξnumber densities are equal. This is true if there is no DMasymmetry which we will assume throughout this work.In the last approximation, we perform a coupling expan-

sion of Eqs. (4.1)–(4.3). After inserting the results of thetwo-point functions into the equation of Gηξ, by using therelations Eqs. (4.9) and (4.10) in the free limit, we obtain forthe four-point correlator to the leading order in gχ :

Gηξðx; y; z; wÞ ≃Gη;0ðx; zÞGξ;0ðy; wÞ þ g2χ

Zx;y∈C

Gη;0ðx; xÞGξ;0ðy; yÞDðx; yÞGη;0ðx; zÞGξ;0ðy; wÞ

þZx;y∈C

Gη;0ðx; zÞGξ;0ðy; yÞð−iÞΣξðy; xÞGξ;0ðx; wÞ þZx;y∈C

Gη;0ðx; xÞGξ;0ðy; wÞð−iÞΣηðx; yÞGη;0ðy; zÞ;

ð4:11Þ


115023-11

where in the last two terms, particle and antiparticle aredisconnected and we have introduced the single-particleself-energies according to

Σηðx; yÞ≡ −ig2χDðx; yÞGη;0ðx; yÞ;Σξðx; yÞ≡ −ig2χDðx; yÞGξ;0ðx; yÞ: ð4:12Þ

The first integral term in Eq. (4.11) contains the ladderdiagram exchange between particle and antiparticle. Sim-ilar equations for Gηη and Gξξ can also be obtained byapplying the same steps. In order to obtain the spectrum ofbound-state solutions as well as a fully nonperturbativetreatment of the Sommerfeld-enhancement we have toresumm Eq. (4.11) somehow.We define our resummation scheme of the four-point

correlator by resumming the ladder exchange, as well as theself-energy contributions in Eq. (4.11) on an equal footing.In other words, we are resumming the leading order terms

in the coupling expansion of the four-point correlator Gηξ

and similar forGηη andGξξ. On the one hand, this is a criticalpoint, since this procedure is not exact and we cannotguarantee that other contributions do not play an importantrole as well, e.g., one of the limitations are systems withlarge coupling constants or large DM density where wecannot decouple the solution of Gηξ from Gηη and Gξξ. Thelatter limitationmight not be a problem sincewhen focussingon the freeze-out the DM density becomes dilute. On theother hand, as we will discuss in detail later in this work, thefinal result based on this resummation scheme behavesphysically, fulfils KMS condition in equilibrium,5

reproduces the correct vacuum limit, enables us to studybound states and DM Sommerfeld-enhanced annihilationat finite temperature, and in some other limits we recoverthe literature results based on linear response theory.Furthermore, a combined resummation of one-particleself-energies and ladder-diagram exchanges seems to benecessary at finite temperature, since similar combinationswould also occurwhen calculating the effective potential fromWilson-loop lines [73] guaranteeing the gauge invariance.Before we can resum Eq. (4.11), we need some further

exact rearrangements and manipulations of the four-pointfunction components. Sincewe are interested in the solutionofGþþ−−

ηξ at equal times, it is sufficient to only consider two-time four-point functions, where we will adopt the shortnotation Gηξðt; t0Þ ¼ Gηξðtxy; t0zwÞ. Certain combinationsof the components of Eq. (4.11) turn out to be closed, e.g., letus define6

GRηξðt; t0Þ≡ θðt − t0Þ½Gþþ−−

ηξ ðt; t0Þ −Gþ−−þηξ ðt; t0Þ

−G−þþ−ηξ ðt; t0Þ þG−−þþ

ηξ ðt; t0Þ�; ð4:13Þ

GAηξðt; t0Þ≡ −θðt0 − tÞ½Gþþ−−

ηξ ðt; t0Þ −Gþ−−þηξ ðt; t0Þ

−G−þþ−ηξ ðt; t0Þ þG−−þþ

ηξ ðt; t0Þ�: ð4:14Þ

In the following we will show that, by using the semigroupproperties of free correlators, the following structure of theretarded equations can be achieved:

GRηξðt; t0Þ ¼ GR

η;0ðt; t0ÞGRξ;0ðt; t0Þ

þ g2χ

Zdt1GR


×Z

dt2ΣRηξðt1; t2ÞGR

η;0ðt2; t0ÞGRξ;0ðt2; t0Þ;

ð4:15Þ

and similar for the advanced. The precise terms contained inthe two-particle self-energy Σηξ will be given later. From theform of Eq. (4.15) it is clear that our resummation scheme asdescribed above is just the replacement of the free two-pointcorrelators at the end byGR

η;0ðt2; t0ÞGRξ;0ðt2; t0Þ → GR

ηξðt2; t0Þ.Let us in the following sketch the way to obtain this

resummable structure. The first term on the r.h.s. inEq. (4.15) can be obtained by using simple relations

Gþ−η;0G

þ−ξ;0 − Gþ−

η;0G−þξ;0 − G−þ

η;0Gþ−ξ;0 þ G−þ

η;0G−þξ;0

¼ Gρη;0G

ρξ;0 ¼ GR

η;0GRξ;0 þGA

η;0GAξ;0: ð4:16Þ

5Another resummation scheme forGηξ we tested can be obtaineddirectly after the steps of the truncation in Eq. (4.8) and Hartree-Fock approximation in Eqs. (4.9)–(4.10). After some algebra, thiswould result in the Bethe-Salpeter equation Gηξðx;y;z;wÞ¼Gηðx;zÞGξðy;wÞþg2χ

Rx;y∈CGηðx;xÞGξðy;yÞDðx; yÞGηξðx; y;z;wÞ.

Note that this equation would be equivalent to the original vacuumBS equation when naively extending the time integration in thelatter equation towards the Keldysh contour C. The main differencehere compared to our resummation scheme is that the two-pointfunctions are fully interacting. The l.h.s. of theþþ −− or− −þþcomponent of this equation fulfils the KMS condition in equilib-rium, when taking the two-time limit. The right-hand side dependsin general on three times. It can be reduced to only two times byassuming a static form for the mediator correlator Dðx; yÞ ¼δCðtx; tyÞVðx − yÞ. Integrating over Keldysh-contour delta func-tion leads to the fact that also the r.h.s. fulfils the KMS condition.However, this is a strong assumption on the form of the mediatorcorrelation function, sending the off-diagonal termsDþ− orD−þ tozero. The simplest possibility to take into account the off-diagonalterms and at the same time obtain a two-time structure also of ther.h.s. would be to assume that every component ofD is proportionalto a time delta function. A crucial observation we have made is thatthis type of approximation seems to violate the KMS conditionthrough the off-diagonal terms, although the r.h.s. has a two-timestructure. The reason might be in the resummation of differentorders in the coupling, as caused by the interacting DM two-pointcorrelators in the BS kernel. The coupling expansion and theresummation of terms of equal order in the coupling parameter (ourscheme), seems to be essential in order to obtain our final BSEq. (4.20), fulfilling the KMS condition.

6This combination is not obvious at first, but when subtractingthe advanced component from the retarded it can be shown thatthe resulting spectral function has a similar completeness relationas the spectral function of two-point correlators [74].


115023-12

In the last step we used GRGA ¼ 0 for equal time productsand when multiplying this with the unit step function as inthe definition Eqs. (4.13) and (4.14) this just projects outthe respective product. The integral term is more compli-cated. Let us consider for simplicity only the last integralterm in Eq. (4.11) and perform the sum over the differentcontributions of the components for the retarded two-timefour-point correlator, resulting in

I ¼Z

dt1dt2GRη;0ðt; t1ÞGR

ξ;0ðt; t0Þð−iÞΣRη ðt1; t2ÞGR

η;0ðt2; t0Þ;

ð4:17Þwhere the retarded one-particle self-energy is defined asΣR ¼ Σþþ − Σþ−. Since all propagators are free, we canuse the semigroup property (see also Appendix A)

GRξ;0ðt; t0Þ ¼ GR

ξ;0ðt; t1ÞGRξ;0ðt1; t2ÞGR

ξ;0ðt2; t0Þ;for t < t1 < t2 < t0: ð4:18Þ

For brevity, we suppress the space integration here. Thisproperty can be used in Eq. (4.17) since the times satisfythe inequality due to the product of retarded correlators,resulting in

I¼Z

dt1GRη;0ðt;t1ÞGR

ξ;0ðt;t1ÞZdt2½ð−iÞΣR

η ðt1;t2ÞGRξ;0ðt1;t2Þ�

×GRη;0ðt2;t0ÞGR

ξ;0ðt2;t0Þ: ð4:19ÞComparing with Eq. (4.15), we indeed find the anticipatedstructure. Applying similar steps to all integral terms inEq. (4.11), we find the following two-time Bethe-Salpeterequations:7

GΦηξðtxy; t0zwÞ ¼ GΦ

ηξðtxy; t0zwÞ þZ

t1;x1;x2

GRηξðtxy; t1x1x2Þ

Zt2;x3;x4

ΣRηξðt1x1x2; t2x3x4ÞGΦ

ηξðt2x3x4; t0zwÞ

þZ

t1;x1;x2

GRηξðtxy; t1x1x2Þ

Zt2;x3;x4

σΦηξðt1x1x2; t2x3x4ÞGAηξðt2x3x4; t0zwÞ

þZ

t1;x1;x2

GΦηξðtxy; t1x1x2Þ

Zt2;x3;x4

ΣAηξðt1x1x2; t2x3x4ÞGA

ηξðt2x3x4; t0zwÞ ð4:20Þ

with Φ¼fþþ−−g;fþ−−þg;f−þþ−g;f−−þþg, and

GR=Aηξ ðtxy; t0zwÞ ¼ GR=A

ηξ ðtxy; t0zwÞ

þZ

t1;x1;x2

GR=Aηξ ðtxy; t1x1x2Þ

×Z

t2;x3;x4

ΣR=Aηξ ðt1x1x2; t2x3x4Þ

×GR=Aηξ ðt2x3x4; t0zwÞ: ð4:21Þ

The products of free correlators are defined as

GRηξðtxy; t0zwÞ≡GR

η;0ðtx; t0zÞGRξ;0ðty; t0wÞ;

and GAηξðtxy; t0zwÞ≡ −GA

η;0ðtx; t0zÞGAξ;0ðty; t0wÞ; ð4:22Þ

and similar for the other components, e.g., Gþþ−−ηξ ¼

Gþ−η;0G

þ−ξ;0 . Furthermore, we introduced the two-particle

self-energies according to

ΣRηξðtxy; t0wzÞ≡ ð−iÞΣR

η ðtx; t0wÞGRξ;0ðty; t0zÞ

þ ð−iÞΣRξ ðty; t0zÞGR

η;0ðtx; t0wÞþ g2χGR

ξ;0ðty; t0zÞD−þðtx; t0zÞGRη;0ðtx; t0wÞ

þ g2χGRη;0ðtx; t0wÞDþ−ðt0w; tyÞGR

ξ;0ðty; t0zÞþ g2χG

þ−ξ;0 ðty; t0zÞDRðtx; t0zÞGR

η;0ðtx; t0wÞþ g2χG

þ−η;0 ðtx; t0wÞDAðt0w; tyÞGR

ξ;0ðty; t0zÞ;ð4:23Þ

where the retarded single-particle self-energy is defined interms of the components of the definition Eq. (5.20),namely ΣR

i ≡ Σþþi − Σþ−

i . The advanced two-particleself-energy is given by

−ΣAηξðtxy;t0wzÞ≡ð−iÞΣA

η ðtx;t0wÞGAξ;0ðty;t0zÞ

þð−iÞΣAξ ðty;t0zÞGA

η;0ðtx;t0wÞþg2χGA

ξ;0ðty;t0zÞD−þðtx;t0zÞGAη;0ðtx;t0wÞ

þg2χGAη;0ðtx;t0wÞDþ−ðt0w;tyÞGA

ξ;0ðty;t0zÞþg2χG

þ−ξ;0 ðty;t0zÞDAðtx;t0zÞGA

η;0ðtx;t0wÞþg2χG

þ−η;0 ðtx;t0wÞDRðt0w;tyÞGA

ξ;0ðty;t0zÞ;ð4:24Þ

7Similar equations where also obtained in [74], although thederivation, particle content, and potential is slightly differentcompared to ours. Nevertheless, further helpful steps for bringingthe integral terms into a resumable form by using semigroupproperties can be found in the Appendix of [74].


115023-13

and the most important statistical components are definedby

σþþ−−ηξ ðtxy; t0wzÞ≡ ð−iÞΣþ−

η ðtx; t0wÞGþ−ξ;0 ðty; t0zÞ

þ ð−iÞΣþ−ξ ðty; t0zÞGþ−

η;0 ðtx; t0wÞþ g2χG

þ−η;0 ðtx; t0wÞD−þðt0w; tyÞGþ−

ξ;0 ðty; t0zÞþ g2χG

þ−η;0 ðtx; t0wÞDþ−ðtx; t0zÞGþ−

ξ;0 ðty; t0zÞ; ð4:25Þ

σ−−þþηξ ðtxy; t0wzÞ≡ ð−iÞΣ−þ

η ðtx; t0wÞG−þξ;0 ðty; t0zÞ

þ ð−iÞΣ−þξ ðty; t0zÞG−þ

η;0 ðtx; t0wÞþ g2χG

−þη;0 ðtx; t0wÞDþ−ðt0w; tyÞG−þ

ξ;0 ðty; t0zÞþ g2χG

−þη;0 ðtx; t0wÞDþ−ðtx; t0zÞG−þ

ξ;0 ðty; t0zÞ: ð4:26Þ

A graphical illustration of the resummation scheme for theretarded or advanced component is shown in Fig. 2. Below,we summarize the essential properties of our two-time BSequations, based on our resummation of one-particle self-energies as well as the mediator exchanges on an equalfooting.

(i) Two-time structure.— The remarkable result of ourresummation scheme is that we achieved a two-timedependence of the Bethe-Salpeter Eqs. (4.20) and(4.21) without assuming a static property of media-tor correlator D or, more general, without assumingany particular form.

(ii) KMS condition.— In general, the two-time correlatorsGþþ−−

ηξ ðt; t0Þ and G−−þþηξ ðt; t0Þ are related in equilib-

rium via the KMS condition: G−−þþηξ ðtxy;t0zwÞ¼

Gþþ−−ηξ ðtþ iβ;xy;t0zwÞ orG−−þþ

ηξ ðtxy;t0 þ iβ;zwÞ¼Gþþ−−

ηξ ðtxy;t0zwÞ. Now, one of the great features ofour resummation scheme is that it respects this KMScondition which is nontrivial since the equations arenot exact. This means that the left- and right-handsides of BS Eq. (4.20) for the componentsGþþ−−

ηξ ðt; t0Þ and G−−þþηξ ðt; t0Þ transform respec-

tively into each other in equilibrium. This can beseen, by Fourier transforming the kernels, neededfor proper analytic continuation of time, and byusing the fact that in equilibrium all quantities onlydepend on the difference of the time variable.Indeed one finds that always the statistical partin the three kernels of Eq. (4.20) transform intotheir counterparts. The solution of our targetcomponent Gþþ−−

ηξ becomes a very simple expres-sion when utilizing the power of the KMS con-dition as we will see in the next section.

(iii) Similar to the two-point functions, two-time retardedand advanced components of the four-point corre-lator are related by complex conjugationGA

ηξðt; t0Þ ¼−½GR

ηξðt0; tÞ�†, as can be shown directly from thedefinition Eqs. (4.13) and (4.14).

(iv) Other components of four-point correlators not listedabove can be constructed by the given ones [74].

(v) The full set of equations are able to describe Bose-Einstein condensation, e.g., relevant for fermionicsystems where bound-state solutions exist and thedensity and chemical potential are in a criticalregime. Since we focus on multi-TeV particles,the density will be always low enough to ignorethose quantum-statistical effects.

V. TWO-PARTICLE SPECTRUM AT FINITETEMPERATURE

For general out-of-equilibrium situations, the coupledsystem of two-time BS Eqs. (4.20)–(4.21) might require afully numerical treatment. However, when relying on somewell-motivated assumptions which are guaranteed forWIMPlike freeze-outs, we show in this chapter that the coupledequations can by drastically simplified. The result will be aformal solution of our target component Gþþ−−

ηξ , appearingin our main number density Eqs. (3.24)–(3.25), in terms ofthe DM two-particle spectral function. Furthermore, wefully provide the details for finding the explicit solution ofthe DM two-particle spectral function from a Schrödinger-like equation with an effective in-medium potential, includ-ing thermal corrections. For a better understanding of thelimitation of this final solution we share, step by step, the

FIG. 2. Resummation scheme shown for the retarded four-point correlator Eq. (4.21). The terms in the brackets belong to one-particleself-energy contributions as well as the mediator exchange between particle and antiparticle. Dots represent terms containing the DMdistribution in the two-particle self-energy. Due to the Boltzmann suppression at the freeze-out, those contributions will be dropped later(see DM dilute limit in Sec. V B).


115023-14

approximations needed. All assumptions leading to thissimple result will be made systematically and discussedseparately in this section.

A. Formal solution in grand canonical ensemble

We assume the DM system to be in a grand canonicalstate where the density matrix takes the form

ρ ¼ 1

Ze−βðH−μηNη−μξNξÞ; H ¼ HNR þMðNη þ NξÞ;

N• ≡Zxj0• ; ð5:1Þ

where • ¼ η, ξ and we assume symmetric DM resulting inequal chemical potentials μ ¼ μη ¼ μξ. Further, we assumethe Hamiltonian to commute with the number operators bytreating the annihilation Γs as a perturbation. This is validas long as the processes driving the system to a grandcanonical state are much more efficient compared toannihilation or the decay of bound states. In this sense,the chemical potential is effectively time dependent. It isrelated to the total number density, appearing on the l.h.s. inEqs. (3.24)–(3.25). The time dependence of the numberdensity is set by the Hubble term and the production andloss terms appearing on the r.h.s.Let us insert the grand canonical density matrix into the

components þþ −− and − −þþ and show how they arerelated. Utilizing the commutation relations of −½Nη;ηð†Þ�¼ð−Þηð†Þ, ½Nξ; ξð†Þ� ¼ ð−Þξð†Þ, one can derive the KMSrelations for the two-time four-point correlators in thepresence of the chemical potentials. Recalling that theHamiltonian is a generator of the time evolution, one findsthe KMS condition for a grand canonical state:

Gþþ−−ηξ;s ðtxy; t0zwÞ ¼ e−2βðM−μÞG−−þþ

ηξ;s ðtxy; t0 þ iβ; zwÞ:ð5:2Þ

Its Fourier transform reads

Gþþ−−ηξ;s ðr; r0;ω;PÞ ¼ e−βðωþ2M−2μÞG−−þþ

ηξ;s ðr; r0;ω;PÞ;ð5:3Þ

where we introduced the Wigner-transformed four-pointcorrelators according to

G••••ηξ;s

�t;X þ r

2;X −

r2; t0;X0 þ r0

2;X0 −

r0

2

�

≡Zω;P

e−iωðt−t0ÞþiP·ðX−X0ÞG••••ηξ;sðr; r0;ω;PÞ: ð5:4Þ

Here, we have used the fact that the operator ρ commuteswith the Hamiltonian H and the translation operator P.Defining the two-particle spectral function as

Gρηξðtxy; t0zwÞ≡G−−þþ

ηξ;s ðtxy; t0zwÞ −Gþþ−−ηξ;s ðtxy; t0zwÞ;

ð5:5Þour target component is formally solved in terms of thespectral function and chemical potential by utilizing theKMS relation for a grand canonical state:

Gþþ−−ηξ;s ðx; x; x; xÞ

¼Zω;P

fBðωþ 2M − 2μÞGρηξð0; 0;ω;PÞ ð5:6Þ

≃Zω;P

e−βðωþ2M−2μÞGρηξð0; 0;ω;PÞ ð5:7Þ

¼ e−2βðM−μÞZ

∞

−∞

d3Pð2πÞ3 e

−βP2=4M

×Z

∞

−∞

dEð2πÞ e

−βEGρηξð0; 0;EÞ: ð5:8Þ

In the second line we approximated the Bose-Einsteindistribution fB as Maxwell-Boltzmann, assuming the DMsystem to be dilute T ≪ ðM − μÞ. The same limit shouldalso be taken in the explicit solution of the spectral function,as is done in the next section. In the last equality (5.8), wehave used that the spectral function only depends on E≡ω − P2=4M (as we will explicitly see later) and adopted theloose notation Gρ

ηξð0; 0;ω;PÞ ¼ Gρηξð0; 0;EÞ. As a conse-

quence of Fourier transformation, the energy integration inEq. (5.8) ranges from minus infinity to plus infinity.If the theory has bound-state solutions in the spectrum,

the two-particle spectral function has strong contributionsat particular negative values of E (binding energy). Thesecontributions are further enhanced by the Boltzmann factore−βE compared to the scattering solutions with positiveenergy, as can be directly seen from Eq. (5.8). This isexpected from the assumption of a grand canonical system.All DM states of energy E must be populated with theBoltzmann factor e−βE for a given DM number density. Asa result, the bound states are preferred compared to thescattering states because their energies are smaller.Under the key assumption of the grand canonical

ensemble, our main number density Eqs. (3.24)–(3.25)are formally closed. This is because the effective chemicalpotential appearing in Eq. (5.8) is related to the totalnumber density by Legendre transformation. On the onehand, the validity of adopting a grand canonical staterequires scattering processes to be efficient in order tokeep DM in kinetic equilibrium with the plasma particlesAμ and ψ . For light mediators this is indeed guaranteed fortimes much later than the freeze-out. On the other hand, ifthe theory has bound-state solutions and is described by agrand canonical ensemble with a single chemical potentialas we have introduced, there appears a hidden assumptionon the internal chemical relation between scattering


115023-15

and bound-state contributions. As we will see later, itautomatically implies the Saha condition for ionizationequilibrium. Ionization equilibrium can only be achievedby efficient radiative processes like the ultrasoft emissionsof mediators. In the description of our theory we havetraced out from the beginning these contributions but cannow formally include them by assuming ionization equi-librium. Thus, a grand canonical description of systemswhere bound states exist is only appropriate if the ioniza-tion equilibrium can be guaranteed. In Sec. VI D, we comeback to this issue in detail, provide an explicit expressionfor the chemical potential and prove the implication ofionization equilibrium.We would like to finally remark that once a grand

canonical picture is appropriate, all finite temperaturecorrections to the annihilation or decay rate are containedin the solution of the two-particle spectral function inEq. (5.8) through Eqs. (3.24) and (3.25). Indeed as we willsee, the negative and positive energy solutions of thespectral function will merge continuously together if finitetemperature effects are strong. In this case it turns out to beimpossible to distinguish bound from scattering statesolutions. Due to the form of Eq. (5.8) it is, however,not required to distinguish between these two contributions.Integrating the spectral function over the whole energyrange automatically takes into account all contributions.In summary, for a grand canonical ensemble, the finitetemperature corrections to Sommerfeld-enhanced annihi-lation and bound-state decay processes are contained in thetwo-particle spectral function and an explicit solution of thelatter quantity is derived in following sections.

B. Two-particle spectral functionin DM dilute limit

A key observation is that we have factored out in (5.8)the leading order contribution in the DM phase-spacedensity and the remaining task is to find the two-particlespectral properties. For the computation of a two-particlespectral function we can now approximate the DM systemto be dilute which is the limit T ≪ ðM − μÞ:

Gþ−η=ξðx; yÞ ≪ G−þ

η=ξðx; yÞ ⇒ G−þη=ξðx; yÞ ≃ Gρ

η=ξðx; yÞ¼ GR

η=ξðx; yÞ −GAη=ξðx; yÞ; ð5:9Þ

where Gρ is the single particle spectral function asintroduced in Sec. II. In the DM dilute limit, the two-particle spectral function is related to the imaginary part ofthe dilute solution of the retarded four-point correlator,where the result is given by:

Gρηξð0; 0;EÞ ≃G−−þþ

ηξ ð0; 0;EÞ¼ Gret

ηξ ð0; 0;EÞ −Gadvηξ ð0; 0;EÞ

¼ 2ℑ½iGretηξ ð0; 0;EÞ�: ð5:10Þ

Here, we definedGretηξ andG

advηξ as the DM dilute limit of the

Eq. (4.21) for GRηξ and G

Aηξ, respectively. For the rest of this

section, the computation of the dilute limit of theseequations is given, proving the claim G−−þþ

ηξ ð0; 0;EÞ ¼2ℑ½iGret

ηξ ð0; 0;EÞ�.Applying the DM dilute limit Eq. (5.9) to the two-time

BS Eq. (4.21) for GRηξ, one finds:

Gretηξ ¼ GR

η;0GRξ;0 þ

ZGR

η;0GRξ;0Σret

ηξGretηξ ; ð5:11Þ

Σretηξ ðtt0xywzÞ ¼ g2χGR

η;0ðtx; t0wÞGRξ;0ðty; t0zÞ

× ½−D−þðtx; t0wÞ −D−þðty; t0zÞþD−þðtx; t0zÞ þDþ−ðt0w; tyÞ�; ð5:12Þ

where Σretηξ is the dilute limit of the two-particle self-energy

ΣRηξ in Eq. (4.23). All space-time dependences remain the

same as in Eq. (4.21) but we suppress them hereafter forsimplicity. Similar limit can be taken for the advancedcomponent. Important to note is that due to the DM dilutelimit, it can be recognized that the retarded Eq. (5.11) andhence the two-particle spectral function are independent ofthe DM number density. For the freeze-out process the DMdilute limit is an excellent approximation. Now to continuewith the proof of G−−þþ

ηξ ð0; 0;EÞ ¼ 2ℑ½iGretηξ ð0; 0;EÞ�, the

equation for the G−−þþηξ [see Eq. (4.20)] in the DM dilute

limit is

G−−þþηξ ¼ GR

η;0GRξ;0 þ GA

η;0GAξ;0

þZ

GRη;0G

Rξ;0Σret

ηξ ½Gretηξ −Gadv

ηξ �

þ GRη;0G

Rξ;0σ

−−þþηξ;dil G

advηξ

þ ½GRη;0G

Rξ;0 þ GA

η;0GAξ;0�Σadv

ηξ Gadvηξ ð5:13Þ

¼ Gretηξ −Gadv

ηξ

þZ

GRη;0G

Rξ;0½−Σret

ηξ þ Σadvηξ þ σ−−þþ

ηξ;dil �Gadvηξ ; ð5:14Þ

where in the step to the last equality we used two-timeBS Eq. (4.21) backward. The statistical two-particle self-energy in the dilute limit is given by

σ−−þþηξ;dil ðtt0xywzÞ¼ g2χ ½GR

η;0ðtx; t0wÞGRξ;0ðty; t0zÞ þ GA

η;0ðtx; t0wÞGAξ;0ðty; t0zÞ�

× ½−D−þðtx; t0wÞ −D−þðty; t0zÞþD−þðtx; t0zÞ þDþ−ðt0w; tyÞ�; ð5:15Þ

where we have used


115023-16

G−þη;0 ðt; t0ÞG−þ

ξ;0 ðt; t0Þ ≃Gρη;0ðt; t0ÞGρ

ξ;0ðt; t0Þ¼ GR


þ GAη;0ðt; t0ÞGA

ξ;0ðt; t0Þ: ð5:16Þ

The first similarity is a consequence of the dilute limit.In the last equality we used GRGA ¼ 0 for equal timeproducts. The integral term in Eq. (5.14) vanishes by notingthat in the dilute limit we have indeed −Σret

ηξ þ Σadvηξ þ

σ−−þþηξ;dil ¼ 0which finally proves the claimG−−þþ

ηξ ð0; 0;EÞ ¼2ℑiGret

ηξð0; 0;EÞ.

C. Retarded equation in static potential limit

In the previous section, we related the two-particle spec-trum to the solution of the retarded equation in the DM dilutelimit: Gρ

ηξð0; 0;EÞ ¼ 2ℑ½iGretηξð0; 0;EÞ�. As a final step, we

further simplify the two-time Bethe-Salpeter Eq. (5.11) forGret

ηξ by taking the proper static limit of themediator correlatorD, resulting finally in a simple Schrödinger-like equation.Westart by acting with the inverse two-particle propagator fromthe left on Eq. (5.11), arriving at

½GRηξ�−1Gret

ηξðtxy; t0zwÞ¼ iδðt − t0Þδ3ðx − zÞδ3ðy − wÞ

þ iZ

t2;x3;x4

Σretηξðtxy; t2x3x4ÞGret

ηξðt2x3x4; t0zwÞ: ð5:17Þ

Here, we suppress spin-indices for simplicity but quote thefinal result in full form later. Let us simplify the interactionkernel in Fourier space, where we take Wigner transform intime τ≡ t − t0:

Gretηξðp1;p2;p0

1;p02;ωÞ

¼Z

d3xd3yd3zd3wdτeiðωτ−p1·x−p2·yþp01·zþp0

2·wÞ

×Gretηξðxyzw; τÞ: ð5:18Þ

Taking this Fourier transform of the kernel leads to twodistinct parts:

dZΣretηξG

retηξ ¼g2χ

Zd3qð2πÞ3 ½I1ðp1;p2;q;ωÞ

×Gretηξðp1;p2;p0

1;p02;ωÞ

þI2ðp1;p2;q;ωÞGretηξðp1−q;p2þq;p0

1;p02;ωÞ�;ð5:19Þ

where I1 results from the sum of the two one-particleself-energy contributions, whereas I2 originates fromthe exchange term between particle and antiparticle(see Fig. 2):

I1 ¼ ð−iÞZ

dω1dω2dω3

ð2πÞ3Gρ

η;0ðω1;p1 − qÞD−þðω3;qÞGρξ;0ðω2;p2Þ þ Gρ

η;0ðω1;p1ÞD−þðω3;qÞGρξ;0ðω2;p2 − qÞ

ω − ω1 − ω2 − ω3 þ iϵ; ð5:20Þ

I2 ¼ iZ

dω1dω2dω3

ð2πÞ3Gρ

η;0ðω1;p1 − qÞD−þðω3;qÞGρξ;0ðω2;p2Þ þ Gρ

η;0ðω1;p1ÞDþ−ð−ω3;qÞGρξ;0ðω2;p2 þ qÞ

ω − ω1 − ω2 − ω3 þ iϵ: ð5:21Þ

We can perform the two ω integrations over one-particlespectral function, where in the free limit they are given by

Gρη;0ðω;pÞ ¼ ð2πÞδ

�ω −

p2

2M

�;

Gρξ;0ðω;pÞ ¼ ð2πÞδ

�ω −

p2

2M

�: ð5:22Þ

Now the integrals are reduced to

I1 ¼ ð−iÞZ

dωð2πÞ

D−þðω;qÞ þDþ−ð−ω;qÞΩ1 − ωþ iϵ

; ð5:23Þ

I2 ¼ iZ

dωð2πÞ

D−þðω;qÞ þDþ−ð−ω;qÞΩ2 − ωþ iϵ

; ð5:24Þ

where Ωi contain the respective on-shell energies. Noticingthat the Fourier transform of Dþþðt; rÞ ¼ θðtÞD−þðt; rÞ þθð−tÞDþ−ðt; rÞ is given by

DþþðΩi;qÞ ¼ iZ

dωð2πÞ

D−þðω;qÞ þDþ−ð−ω;qÞΩi − ωþ iϵ

;

ð5:25Þ

we can now take the proper static limit Ωi → 0 whichresults in


115023-17

idZ

ΣretηξG

retηξ

¼ ð−iÞg2χZ

d3qð2πÞ3 ½D

þþð0;qÞGretηξ ðp1;p2;p0

1;p02;ωÞ

−Dþþð0;qÞGretηξ ðp1 − q;p2 þ q;p0

1;p02;ωÞ�: ð5:26Þ

Introducing Wigner-momenta and Fourier transformingback with respect to the difference variables

P ¼ ðp1 þ p2Þ − ðp01 þ p0

2Þ;p ¼ ðp1 − p2Þ=2; p0 ¼ ðp0

1 − p02Þ=2; ð5:27Þ

Gretηξ ðr;r0;P;ωÞ ¼

Zd3pð2πÞ3

d3p0

ð2πÞ3 eiðp·r−p0·r0ÞGret

ηξ ðp;p0;P;ωÞ;

ð5:28Þwe finally end up with the Schrödinger-like equation for theretarded four-point correlator in the static limit of thepotential:�ω −

P2

4Mþ ∇2

r

Mþ iϵ − VeffðrÞ

�Gret

ηξ ðr; r0;P;ωÞ

¼ Tr½12×2�iδ3ðr − r0Þ: ð5:29Þ

Now we see that the retarded and hence also the spectralfunction only depends on E≡ ω − P2=ð4MÞ.The retarded two-time BS equation in static and dilute

limit can be written as:�∇2r

MþEþ iϵ−VeffðrÞ

�Gret

ηξ ðr;r0;EÞ ¼ Tr½12×2�iδ3ðr− r0Þ;

ð5:30Þwhere the effective in-medium potential is defined as

VeffðrÞ≡ −ig2χZþ∞

−∞

d3qð2πÞ3 ð1 − eiq·rÞDþþð0;qÞ: ð5:31Þ

The spectral function, we would actually like to com-pute, is obtained from the solution of this equationaccording to the relation Gρ

ηξð0;0;EÞ¼2ℑ½iGretηξ ð0;0;EÞ�,

as proven in the previous section. In the next section, wewill derive the explicit solution of the retarded equationwhere we will further approximate the static mediatorcorrelator Dþþ in the hard thermal loop limit, as alreadygiven in Eq. (3.11). The first term in the effective potentialin Eq. (5.31) originates from the sum of the two single-particle self-energies, while the second term accounts forthe mediator exchange between particle and antiparticle.The trace in the Schrödinger-like Eq. (5.30) takes thecorrect spin summation into account, which we havesuppressed in this section for simplicity.

D. Explicit solution in static HTL approximation

Taking the static HTL approximation of the masslessmediator as derived in Eq. (3.11), the effective potentialaccording to Eq. (5.31) results in

VeffðrÞ ¼ −αχmD −αχre−mDr − iαχTϕðmDrÞ; ð5:32Þ

ϕðxÞ ¼ 2

Z∞

0

dzz

ðzþ 1Þ2�1 −

sinðzxÞzx

�; ð5:33Þ

and ϕð0Þ ¼ 0 and ϕð∞Þ ¼ 1. One can recognize the realpart of the potential is corrected by the Debye mass asexpected. It shifts the energy by αχmD (twice the Salpetercorrection of single particle self-energies) and screensthe Coulomb potential. At the same time, the effectivepotential contains an imaginary part. The physical meaningof this term is scatterings of DM with light particles in thethermal plasma. If particle and antiparticle are far away,the imaginary part must be solely determined by scatteringwith the thermal plasma without the Yukawa force. Onecan also see that this is actually the case, since theimaginary part becomes twice the thermal width of singleparticles −iαχT for r → ∞ (see single particle correctionsin Appendix E 2, as well as Salpeter correction inAppendix E 3). This property follows from our resumma-tion scheme, treating the DM self-energy on an equalfooting with the ladder diagram exchange. Since the finitetemperature corrections introduce a constant imaginaryterm for large distances, we can drop the iϵ term in thefollowing derivation of the explicit solution of Eq. (5.30).This solution will be general and contains also the correctvacuum limit, where iϵ has to be carefully taken intoaccount. The effective potential in Eq. (5.32) was firstobtained in [73] and reproduced subsequently by othermethods [70,75]. Let us remark that we derived it inde-pendently, i.e., for the first time starting from a set of two-time Bethe-Salpeter equations on the Keldysh contour.To derive the solution, we expand Gret

ηξ in terms of partialwaves

Gretηξ ðr; r0;EÞ ¼

Xl

2lþ 1

4πPlðcos θr;r0 Þð−iÞGret

ηξ;lðr; r0;EÞ;

ð5:34Þ

leading to the l ¼ 0 (s-wave) equation:�−

1

M1

r2∂rðr2∂rÞ − Eþ VeffðrÞ

�Gret

ηξ;sðr; r0;EÞ

¼ Tr½12×2�1

rr0δðr − r0Þ: ð5:35Þ

The physically relevant boundary conditions we impose onGret

ηξ;s are listed below.


115023-18

(i) Gretηξ;sðr; r0;EÞ is finite ∀ r, r0.

(ii) For jr − r0j → ∞, Gretηξ;s decays exponentially (as a

consequence of constant imaginary potential).Note here that, since we are working in the dilute limit, theFeynman propagator and the retarded function are thesame. These requirements set the form of the solutionuniquely (see also [46,76]):

Gretηξ;sðr; r0;EÞ ¼ Tr½12×2�

Mrr0

½θðr − r0Þg>ðrÞg<ðr0Þþ θðr0 − rÞg<ðrÞg>ðr0Þ�; ð5:36Þ

where g>=< are the solutions of the homogeneous differ-ential equation�

−1

M∂2r − Eþ VeffðrÞ

�g>=<ðrÞ ¼ 0; ð5:37Þ

whose boundary conditions are given as follows:(i) g<ð0Þ ¼ 0 and g0<ð0Þ ¼ 1;(ii) g>ð0Þ ¼ 1 and g>ðrÞ decays exponentially for

r → ∞.One can explicitly check that the solution of the form (5.36)with the boundary conditions for g>=< fulfils the require-ments on Gret

ηξ;s as listed above.The two-particle spectral function, needed to compute

the annihilation or decay rate according to Eqs. (5.8)and (5.10), can be written in terms of the homogeneoussolution as

Gρηξð0; 0;EÞjl¼0 ¼ 2ℑ

hiGret

ηξ ð0; 0;EÞ��l¼0

i¼ 1

2πℑhlimr;r0→0

Gretηξ;sðr; r0;EÞ

i¼ 1

2πTr½12×2�Mℑ½g0>ð0Þ�: ð5:38Þ

In the last term, prime stands for the derivative with respectto r. Formally this solution is correct but might betroublesome in the numerical evaluation, since the realpart of g0>ð0Þ has a singularity due to the 1=r behavior ofthe effective potential. To resolve this issue, we rewrite theimaginary part of g0>ð0Þ by means of a different solution.We closely follow the discussion given in Ref. [76]. Letus define another singular solution gs whose boundaryconditions are given by gsð0Þ ¼ 1 and ℑ½g0sð0Þ� ¼ 0. Theoutgoing solution for r → ∞ can be expressed by a linearcombination of

g> ¼ gs þ Bg<: ð5:39Þ

By definition, we have ℑ½g0>ð0Þ� ¼ ℑ½B�. The fact that g> isoutgoing forces it to decay for r → ∞, which determines B.And thus, one finds

ℑ½g0>ð0Þ� ¼ ℑ½B� ¼ −ℑ�limr→∞

�gsðrÞg<ðrÞ

��: ð5:40Þ

The physical quantity ℑ½B� we would like to compute doesnot depend on the choice of ℜ½g0sð0Þ�. This is why we canget the correct result without handling the divergence ofℜ½g0sð0Þ�. The final step is to rewrite the singular solution as

gsðrÞ ¼ −g<ðrÞZ

r

0

dr01

g2<ðr0Þ: ð5:41Þ

One may also write down the expression for g> by usingEq. (5.41):

g>ðrÞ ¼ g<ðrÞZ

∞

rdr0

1

g2<ðr0Þ: ð5:42Þ

One can check that it fulfills the boundary conditionsgsð0Þ ¼ g>ð0Þ ¼ limr→0rð1=rÞ ¼ 1 and ℑ½g0sð0Þ� ¼ℑ½g>ð0Þ� − ℑ½B� ¼ 0. Plugging Eq. (5.41) into Eq. (5.40)and recalling the relation Eq. (5.38), we finally arrive atthe convenient form to evaluate the imaginary part (seealso [64]):

ℑ½Gretηξ;sð0; 0;EÞ� ¼ Tr½12×2�Mℑ½g0>ð0Þ�

¼ Tr½12×2�Mlimδ→0

Z∞

δdrℑ

�1

g<ðrÞ�

2

:

ð5:43Þ

The great advantage of this general solution is that itapplies for the whole two-particle energy spectrum of ourtheory, i.e., for negative and positive E. Here, we haveintroduced the tolerance δ ≪ 1 as initial value for numericalstudies. Let us finally introduce dimensionless variables,expressing distances in terms of the Bohr radius x≡ αχMr,and summarize the equations in terms of these units. Thes-wave part of two-particle spectral function reads:

Gρηξð0; 0;EÞjl¼0

¼ 1

2πTr½12×2�αχM2 lim

x0→0

Z∞

x0dxℑ

�1

g<ðxÞg<ðxÞ�: ð5:44Þ

The homogeneous equations for a massless and massivemediator are given by:

g00<ðxÞ þ�

Eα2χM

þ mD

αχMþ 1

xe−

mDαχM

x þ iT

αχMϕ

�mD

αχMx

��g<ðxÞ ¼ 0; ð5:45Þ


115023-19

g00<ðxÞ þ�

Eα2χM

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2

V þm2D

p−mV

αχMþ 1

xe−

ffiffiffiffiffiffiffiffiffiffim2Vþm2

D

pαχM

x þ iT

αχM1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þm2V=m

2D

p ϕ

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2

V þm2D

pαχM

x

��g<ðxÞ ¼ 0: ð5:46Þ

In practice, we use the following initial conditions whichcan be obtained by power series approach:

g<ðxÞ ¼ x − x2=2þ iγix5; ð5:47Þ

γC ¼ −1

40

TαχM

�mD

αχM

�2

; ð5:48Þ

γY ¼−1

40

TαχM

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þðmV=mDÞ2

p � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2

Dþm2V

pαχM

�2

; ð5:49Þ

where γC applies for the Coulomb case Eq. (5.45) while γYcan be taken for the Yukawa case Eq. (5.46). To efficientlydeal with the sometimes highly oscillatory integrand inEq. (5.44), local adaptive integrating methods are useful.In finding the initial power in x for the imaginary part wehave assumed that ϕðxÞ ∼ 1

2x2 for small x [64] which is

only approximately true. In general, one has to carefullycheck if the correct value of the potential is close enoughto this approximation at the initial value which we havedone for the numerical results presented in subsequentsections.

VI. DMNUMBER DENSITY EQUATION IN GRANDCANONICAL ENSEMBLE

In the previous section, we have obtained a formalsolution of the out-of-equilibrium term Gþþ−−

ηξ;s , enteringour main number density Eq. (3.24). This was achieved byassuming the DM system is in a grand canonical state,formally solving Gþþ−−

ηξ;s in terms of the chemical potentialand two-particle spectral function by KMS relation.Inserting this formal solution given in Eq. (5.8) into themain number density Eq. (3.24) results in our masterformula for the DM number density equation in a grandcanonical ensemble:

_nη þ 3Hnη ¼ −2ðσvrelÞGþþ−−ηξ;s ðx; x; x; xÞjeqðeβ2μη½nη� − 1Þ;

ð6:1Þ

where a symmetric plasma 2μη ¼ μη þ μξ is assumed and(σvrel) is the s-wave tree-level annihilation cross section.The latter quantity is averaged over initial internal d.o.f.(spin) and summed over final.The chemical potential μη½nη� is a function of the total

number density nη as it appears on the left-hand side of ourmaster formula. The term Gþþ−−

ηξ;s jeq is the chemicalequilibrium limit μ → 0 of Eq. (5.8), given by

Gþþ−−ηξ;s ðx; x; x; xÞjeq ¼ e−β2M

Z∞

−∞

d3Pð2πÞ3 e

−βP2=4M

×Z

∞

−∞

dEð2πÞ e

−βEGρηξð0; 0;EÞjl¼0:

ð6:2Þ

We presented a general method in the previous sectionof how to compute the in-medium two-particle spectralfunction Gρ

ηξ explicitly. It contains finite temperaturecorrections to the Sommerfeld enhancement and bound-state decay. The only parameter left in Eq. (6.1) is thechemical potential, which has not yet been explicitlysolved. The chemical potential μη½nη� can be obtained intwo steps as demonstrated in the following. First, the totalnumber density as a function of the chemical potential iscomputed. For a grand canonical ensemble this followsfrom basic relations of quantum statistical mechanics and isgiven by

nη½μη� ¼∂p∂μη

��T;Ω

; ð6:3Þ

where the total pressure is

pΩ ¼ T lnZgrðΩ; T; μÞ ¼ T ln Tr½e−βðH−μηNη−μξNξÞ�: ð6:4Þ

Here, Ω is the volume and Zgr is the grand canonicalpartition function. Second, by inversion of Eq. (6.3) oneobtains the chemical potential as a function of the totalnumber density. The functional dependence of μ½nη� on thetotal number density can be nontrivial especially for thecase if bound-state solutions exist as we will see later.The nη on the l.h.s. of Eq. (6.3) is equivalent to nηappearing on the l.h.s. of our master Eq. (6.1).In subsequent sections of this chapter we demonstrate

how powerful our master Eq. (6.1) is. We self-consistentlycompute the component Gþþ−−

ηξ;s ðx; x; x; xÞjeq and thechemical potential μ. This means in both terms the sameapproximations should be made to obtain a well-behavednumber density equation.For a better understanding, we would like to start in the

next Sec. VI A with the simplest case of our theory bytaking the zero self-interaction and zero finite temperaturecorrection limit. This means we take gχ → 0 and gψ → 0 inthe effective in-medium potential Eq. (5.32) and computethe spectral function. The same limits are applied to theHamiltonian entering in Eq. (6.4) to compute the totalpressure. Under these limits, our master Eq. (6.1) reduces to


115023-20

the conventional Lee-Weinberg equation, describing con-stant s-wave annihilation of DM.As a next step, we allow for long-range self-interactions

but neglect finite temperature corrections. This correspondsto the limit gψ → 0, leading to the fact that the remainingterm in the effective in-medium potential is the standardCoulomb or unscreened Yukawa potential. In Sec. VI B, thetwo-particle spectrum for this simple case of our theory isshown. We stress the point that only in this limit there is adirect relation between spectral function and standardexpressions for the Sommerfeld-enhancement factor orthe decay width of the bound states. Section VI C com-pletes the results by computing the chemical potential forthe same limit. Combining the analytic expressions for thespectral function and chemical potential, we prove ourmaster formula Eq. (6.1) to be consistent with the classicalon-shell Boltzmann equation treatment for vanishing ther-mal corrections. We also point out that adopting a grandcanonical ensemble with one single time-dependent chemi-cal potential as in our master formula implies ionizationequilibrium between the scattering and bound states. Adetailed discussion is given on the validity of ionizationequilibrium during the freeze-out process. If no bound-statesolutions exist, the only limitation of our master formula isessentially kinetic equilibrium [77,78].We relax the assumption of zero finite temperature

corrections in Sec. VI D. This brings us to another centralresult of this work: a DM number density equation,generalizing the conventional Lee-Weinberg equationand classical on-shell Boltzmann equation treatment as aconsequence of accounting simultaneously for DM anni-hilation and bound-state decay at finite temperature.However, it should be noted that this equation inSec. VI D strictly speaking only applies to the narrowthermal width case and is therefore less general comparedto our master Eq. (6.1). This means we have neglectedin Sec. VI D imaginary-part corrections to the effectivein-medium potential for the computation of thechemical potential. While we can fully account forthese non-Hermite corrections in the computation ofGþþ−−

ηξ;s ðx; x; x; xÞjeq, it remains an open question of thiswork of how to consistently compute the chemical potentialfor the broad thermal width case. The broad thermal widthcase for Gþþ−−

ηξ;s ðx; x; x; xÞjeq we compute numerically laterin this work (see Sec. VII). Nevertheless, we demonstratethat the chemical potential and the two-particle spectralfunction entering the number density equation in Sec. VI Dcan be evaluated self-consistently in the narrow thermalwidth limit. This approach, taking leading finite temper-ature real-part corrections into account, is already moregeneral of what has been computed so far in the literature.In principle, it is possible to take a nonconsistent approachand computeGþþ−−

ηξ;s ðx; x; x; xÞjeq including imaginary partsin the potential while only including real-part corrections tothe chemical potential. However, some care must be taken

when doing so. This is because the chemical potentialcorrects the functional form of the number density depend-ence in our master equation. We discuss in more detail thepossibility of taking a non-self-consistent approach by theend of Sec. VI D.Finally in Sec. VI E, we compare our master Eq. (6.1) to

the previous literature, relying on the method of linearresponse theory. Consistency is proven in the linear regimeclose to chemical equilibrium.

A. Recovering the Lee-Weinberg equation

We take the limit of zero self-interactions αχ → 0while keeping the annihilation term Γs as a perturbation.It should be emphasized again that we have to approximatethe spectral function and the chemical potential both in thesame limit in order to obtain a self-consistent solution. Thefree spectral function without self-interactions and the idealpressure are given by

Gρηξð0; 0;EÞjl¼0 ¼ θðEÞ 1

2πTr½12×2�M3=2E1=2; ð6:5Þ

p0Ω ¼ T ln Tr½e−βðH0−μηNη−μξNξÞ�: ð6:6Þ

Here,H0 is the free Hamiltonian and for a derivation of thisresult for the two-particle spectral function directly startingfrom the general expression Eq. (5.44) can be found inAppendix D. The number density can be obtained fromEq. (6.3) by using the ideal pressure:

nη½μη� ¼∂p0

∂μη��T;Ω

¼ Gþ−η;0 ðx; xÞ

¼ eβμ2Z

d3pð2πÞ3 e

−βðMþp2=2MÞ ¼ eβμneqη;0; ð6:7Þ

neqη;0 ¼ 2

�MT2π

�3=2

e−βM: ð6:8Þ

In the second equality of the first line, we find the relationbetween ideal number density and the noninteractingcorrelatorGþ−

η;0 ðx; xÞ. The latter quantity has to be evaluatedin a grand canonical ensemble, which we have done in thethird equality by using KMS condition and the DM dilutelimit (see Appendix E 1). The DM dilute limit should betaken in the computation of Gþ−

η;0 ðx; xÞ to be consistent withthe computation of the spectral function. For the latterquantity we have seen in Sec. V B only in the DM dilutelimit it is independent of the DM number density and ourgeneral solution Eq. (5.44) relies on this assumption. In thelast equality of the first line we defined the conventionalchemical equilibrium number density of ideal particles.Finally, we obtain from the last equality the noninteracting(ideal) chemical potential by inversion: βμ ¼ ln½nη=neqη;0�.Note that this inversion can only be done analytically if one


115023-21

approximates the Fermi-Dirac distribution as Maxwell-Boltzmann (which is our DM dilute limit). Entering theseresults of the spectral function and the chemical potentialinto our master formula for the DM number densityEq. (6.1), leads to the conventional Lee-Weinberg equationfor DM particles with zero self-interactions:

_nη þ 3Hnη ¼ −hσvreli½n2η − ðneqη;0Þ2�: ð6:9Þ

Here, we have recovered the standard thermal averagedcross section by using the simple substitution E ¼ Mv2rel=4for the positive energy spectrum:

hσvreli ¼ðM=TÞ3=2

2ffiffiffiπ

pZ

∞

0

dvrele−v2rel

M

4T v2relðσvrelÞ ð6:10Þ

¼ ðσvrelÞ: ð6:11Þ

The last equality holds for constant s-wave annihilationcross section (σvrel) as it is the case for our model.

B. Spectral function, Sommerfeld enhancement factor,and decay width for vanishing thermal corrections

We turn now to the interacting case αχ ≠ 0 and computethe two-particle spectral function. The s-wave two-particlespectral function is numerically solved according toEq. (5.44) in the limit of vanishing finite temperaturecorrections and the results are shown in Fig. 3. Poles in thenegative energy spectrum represent the bound states, whilethe spectrum is continuous for the scattering states atpositive energy. In the vacuum limit one clearly sees thatthe scattering states can be separated from the bound-statecontribution at E ¼ 0. Due to this separation, the solutionof the spectral function is directly related to theSommerfeld enhancement factor SðvrelÞ and bound-statedecay width Γn. These relations are given below and now itbecomes clear that the two-particle spectral function asshown in Fig. 3 is, for the vacuum case, just a convenientway of presenting all contributions simultaneously.The relations between two-particle spectral functionGρ

ηξ,Sommerfeld enhancement factor S, and decay width Γn inthe limit of vanishing finite temperature corrections aregiven by:

ðσvrelÞGρηξð0; 0;EÞjE>0;l¼0

¼ 1

4πTr½12×2�M2vrelðσvrelÞSðvrelÞ; ð6:12Þ

ðσvrelÞGρηξð0; 0;EÞjE<0;l¼0

¼ π

2Tr½12×2�

Xn

δðE − EBnÞΓn; ð6:13Þ

where E ¼ Mv2rel=4 for the scattering states, EBnis the

(negative) binding energy for the bound states, and

(σvrel) is the tree-level s-wave annihilation cross section.Γn is decay width of the bound state (not to be confusedwith our annihilation term Γs at the beginning ofthis work).These relations can be proven directly from our general

solution Eq. (5.44), see Appendix D for a derivation. On afirst look, the spectral function in the vacuum case seemsjust a nice way of presentation. Instead one shouldemphasize that the notion of spectral function is moregeneral and unifies the picture of scattering state annihi-lation and bound-state decay. This observation becomesimportant for the finite temperature case discussed inSec. VII, where it is impossible to separate or distinguishbetween annihilation and decay. The spectrum includesboth. Only in the absolute vacuum case a clear distinctionbetween annihilation and decay can be made.Coming back to Fig. 3 and now keeping in mind the

relations Eqs. (6.12)–(6.13). There is an infinite number ofexited S-bound states for the Coulomb case (left plot) withbinding energy EBn

¼ −α2χM=ð4n2Þ, where n is the numberof the exited states and n ¼ 1 is the ground state withlowest binding energy, shown as the pole most to the left.At small positive energies, where vrel ≲ αχ , the spectralfunction is constant, resulting in the familiar scalingSðvrelÞ ∝ v−1rel according to Eq. (6.12).In the Yukawa potential case, shown in the right plot

of Fig. 3, there is a finite number of bound-state solutions.For certain ratios of ϵϕ ≡mV=ðαχMÞ there exist a bound-state solution with zero binding energy (E ¼ 0). For thosespecial cases the Sommerfeld enhancement factor scales asSðvrelÞ ∝ v−2rel for vrel ≲mV=M ≪ αχ, called on-resonanceregime, leading to an interesting observational impact oncosmology at very late times [79,80]. Roughly, those poleswhere the spectral function would diverge are at the on-resonance condition ϵϕ ¼ 6=ðm2π2Þ where m is integer.8

The on-resonance divergences give rise to partial waveunitarity violation of the total cross section, as can beseen in the right plot of Fig. 3 at E ¼ 0. It has been pointedout in [54] that once the imaginary contribution of theannihilation part (proportional to our Γs) is included self-consistently in the solution of the Schrödinger-likeEq. (5.30), then the Sommerfeld enhancement starts tosaturate below the unitarity limit.9 This means that forsome small velocity there is a transition from the divergent

8This is exactly true for the Hulthen potential case which isanalytically solvable, while for the Yukawa case this resonancecondition slightly deviates.

9Since we have treated the annihilation term Γs as a perturba-tion, where the leading order contribution was found to contributeto the change in the number density, the contribution of Γsdoes not occur in Eq. (5.30). We will see later that at finitetemperature the imaginary parts in the potential will dominateover the imaginary contribution from the annihilation term andthus at sufficiently finite temperature we will always get asaturation below the unitarity bound.


115023-22

scaling SðvrelÞ ∝ v−2rel to SðvrelÞ ∝ const which results inzero spectrum at zero energy. The saturation is alwayspresent if the on-resonance condition is not exactly ful-filled. The other extreme case is if ϵϕ is taken exactly inbetween neighboring on-resonance values, called off-resonance. Then, SðvrelÞ never scales stronger thanSðvrelÞ ∝ v−1rel and at some small velocity of the order vrel ≲mV=M the Sommerfeld enhancement factor starts tosaturate and the spectral function approaches zero.

C. DM number density equation for vanishingthermal corrections

In the previous section we have proven in the limit ofvanishing finite temperature corrections a relation betweenspectral function, standard expression of Sommerfeldenhancement factor and the bound-state decay width.Inserting these relations Eqs. (6.12)–(6.13) into our masterEq. (6.1) leads to the following differential equation for thetotal number density nη:

_nη þ 3Hnη ¼ −hσvreli½ðneqη;0Þ2eβ2μ − ðneqη;0Þ2�−Xi

Γi½neqBi;0eβ2μ − neqBi;0

�: ð6:14Þ

In the limit of zero chemical potential, defining chemicalequilibrium, the r.h.s. vanishes as expected. Here, werecovered the thermal averaged Sommerfeld enhancementfactor:

hσvreli ¼ðM=TÞ3=2

2ffiffiffiπ

pZ

∞

0

dvrele−v2rel

M

4T v2relðσvrelÞSðvrelÞ:

ð6:15Þ

The chemical equilibrium number density for the scatteringstates neqη;0 was coming out as already defined in Eq. (6.8).

This outcome is fully consistent with the result one wouldget from integrating the Maxwell-Boltzmann equilibriumphase-space density (multiplied by spin factor 2) of non-relativistic particles. The chemical equilibrium numberdensity of the bound-states was defined as

neqBi;0¼

�2MT2π

�3=2

e−βMBi ; ð6:16Þ

where the mass of bound-state i is MBi¼ 2M − jEi

Bj andthe subscript 0 stands for ideal bound states, respectively.The term in front of the exponential in the bound-statenumber density Eq. (6.16) needs some further explanation.Since we have only considered s-wave contributions to thespectral function, neqBi;0

is the equilibrium number density ofthe ith exited para-WIMPonium. The decay, as well as theannihilation, of ortho-WIMPonium into three Aμ would bea p-wave process. To form para-WIMPonium there is onlyone spin option while for ortho-WIMPonium there are 3,consistent with the picture of having in total 4 spin d.o.f.Therefore, the spin factor 1 in Eq. (6.16) comes outcorrectly. When carefully looking at the term in front ofthe exponential in Eq. (6.16), it can be seen that thenormalization of the distribution came out as like integrat-ing the phase space density:

neqBi;0¼

Zd3Pð2πÞ3 e

−βðMBiþP=4MÞ: ð6:17Þ

The kinetic term P=4M of the bound state misses thecorrection coming from the binding energy, because theconventional normalization would give ðneqBi;0

ÞðcÞ ¼ðMBi

=2πÞ3=2e−βMBi . The reason why this correction of theorder OðEi

B=MÞ does not come out as in the conventionalcase can be explained by how we have approximated the

FIG. 3. S-wave two-particle spectral function vs the energy E in units of typical freeze-out temperature shown for a standard Coulomb(left) and Yukawa (right) potential. The two-particle spectral function enters directly our master formula and is weighted by theBoltzmann factor for all the energy range.


115023-23

Martin-Schwinger hierarchy. In our paper, we expand theequation for the four-point correlator around the product oftwo free propagators. This is why the spectral function onlydepends onE ¼ ω − P=4M as we have shown in Eq. (5.29).If we would iterate the solution, e.g., correcting the freecorrelators and inserting them again into the solution ofthe four-point correlator, we could obtain the conventionalresult. However, note that the correction is small forperturbative systems, since typically OðEi

B=MÞ ∼Oðα2Þ.We are turning to the discussion of the chemical potential

μ in Eq. (6.14). The chemical potential can be obtained byinverting the number density as a function of the chemicalpotential. For an ideal gas description it is known that thetotal number density of η-particles nη, as it appears on thel.h.s. of Eq. (6.14), is in general just given by the sum ofscattering and bound-state contributions:

nη ¼ nη;0 þXi

nBi;0 ð6:18Þ

nη;0 ¼ neqη;0eβμη ; ð6:19Þ

nξ;0 ¼ neqξ;0eβμξ ; ð6:20Þ

nBi;0 ¼ neqBi;0eβμBi : ð6:21Þ

Since we have imposed a grand canonical state with onlyone single chemical potential, the chemical potentials forthe scattering and bound states are related and therefore thenumber densities are not independent quantities. Assuminga grand canonical ensemble with only one time dependentchemical potential μ implies 2μ ¼ 2μη ¼ 2μξ ¼ μBi

, whichleads to the relation

nBi;0 ¼ 2

�π

MT

�3=2

n2η;0e−βjEi

Bj: ð6:22Þ

This is nothing but the Saha ionization equilibrium con-dition. To see it explicitly, let us insert Eq. (6.22) intoEq. (6.18), leading to a quadratic equation for the numberdensity of free scattering states:

nη ¼ nη;0 þ KidðTÞnη;0nη;0; KidðTÞ ¼P

inBi;0

nη;0nη;0: ð6:23Þ

KidðTÞ is according to Eq. (6.22) independent of thechemical potential. This quadratic Eq. (6.23) can be solved,leading to the degree of ionization αid for ideal gases:

nη;0nη

¼ αidðnηKidðTÞÞ; αidðxÞ ¼ 1

2xð ffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ 4xp

− 1Þ:

ð6:24Þ

The chemical potential can now be obtained from thisequation by using nη;0 ¼ neqη;0e

βμ resulting in

βμ ¼ ln

�αidnηneqη;0

�: ð6:25Þ

Inserting this chemical potential into Eq. (6.14) we finallyend up with the Boltzmann equation for vanishing finitetemperature corrections, ideal gas approximation, and thesystem in a grand canonical state:

_nη þ 3Hnη ¼ −hσvreli½ðαidnηÞ2 − ðneqη;0Þ2�

−Xi

ΓineqBi;0

��αidnηneqη;0

�2

− 1

�: ð6:26Þ

The equation is closed in terms of the total numberdensity nη. Before we discuss this result in detail, let usconsider the case where we would have treated all boundand scattering states to be independent. This could havebeen realized in Eq. (6.14) by assigning different chemicalpotentials to scattering and bound states. Then we wouldhave ended up with decoupled equations:

_nη;0 þ 3Hnη;0 ¼ −hσvreli½n2η;0 − ðneqη;0Þ2�;Xi

_nBi;0 þ 3HnBi;0 ¼ −Xi

Γi½nBi;0 − neqBi;0�: ð6:27Þ

These are the standard equations if bound and scatteringstates are decoupled. They might be helpful to understandEq. (6.26) better. Namely, when adding the well-knownBoltzmann Eqs. (6.27) and imposing ionization equilib-rium, one would end up with Eq. (6.26). We summarize anddiscuss the main findings of this section below.(i) The differential Eq. (6.26) describes the out-of-

chemical equilibrium evolution of the total number density,including the reactions ηξ⇌ fAA;ψψg (Sommerfeld-enhanced annihilation and production) and ðηξÞB ⇌fAA;ψψg (bound-state decay and production) under theconstraint of ionization equilibrium for all times. The totalnumber density nη counts both: free particles as well asparticles in the bound state. Equation (6.26) is equivalent tothe coupled set of Boltzmann equations including softemissions and absorptions [21] in the limit of ionizationequilibrium. The equations are independent of the bound-state formation or ionization cross section since the ratesare, by assumption, balanced. One can also see it from adifferent perspective. Via Eq. (6.26) it is very elegant toinclude bound-state formation or dissociation processeswithout calculating the cross sections or solving a coupledsystem of differential equations. Note that when comparingour single equation to the coupled set of Boltzmannequations in Refs. [21], the last term in Eq. (6.26) account-ing for the direct production of bound states from two


115023-24

photon annihilation (inverse decay) was dropped. It mighthave little impact since bound-state formation from radi-ative processes can be much more efficient around thefreeze-out.(ii) Let us discuss some asymptotic regimes of

Eq. (6.26), assuming the system has bound-state contribu-tions. For this case, the ionization degree αid has a non-trivial dependence on the total number density nη, as can beseen from Eq. (6.24). This leads to the fact that the collisionterm of the number density Eq. (6.26) has in general neithera linear nor a quadratic dependence on nη, as it is for thedecoupled conventional Eqs. (6.27). However, we recovercorrectly the quadratic and linear form in some regimesdiscussed now. Close to the freeze-out, the temperature ismuch larger than the binding energy of bound states. As aconsequence, the bound-state contribution in the ionizationdegree is subdominant and can be neglected. This can beseen directly from Eq. (6.23). In the high temperatureregime x ≪ 1, leading to αid ∼ 1 (fully ionized DMplasma), the dominant part of the r.h.s. of the numberdensity Eq. (6.26) is quadratic in nη. At late times, roughlywhen temperature becomes of the order the binding energy,the contribution of the bound states becomes stronglyenhanced due to Boltzmann factor ∝ eβjEBj in KidðTÞ.For this low temperature regime x ≫ 1, leading to

αid ≃ 1=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinηKidðTÞ

pwhich is much smaller than unity.

This behavior is expected since bound states with energy2M − jEi

Bj are thermodynamically more favored comparedto the scattering states with energy 2M for T < jEi

Bj.Therefore, the equilibrium limit for low temperature isthat most of the particles are contained in the lowestbound state (assuming for a moment that such state is

stable). Inserting the low temperature behavior of αid ≃1=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinηKidðTÞ

pinto Eq. (6.26) results in the fact that at late

times the r.h.s. the of number density equation is linear innη and the proportionality factor is effectively the bound-state decay rate. Since typically the decay rates are muchlarger compared to the Hubble rate and the number densityequation is linear in nη, the total number density getsdepleted exponentially fast in time. This means at latetimes, if ionization equilibrium is assumed, also the freeDM particle number decreases simply because it mustfollow the exponential fast decay of the bound states inorder to maintain the imposed ionization equilibrium. Insummary, if bound-state solutions are present and onewould integrate numerically Eq. (6.26) until today,effectively no DM would remain as a consequence ofthe imposed ionization equilibrium. However, we knowfrom the full coupled set of classical Boltzmannequations [21] that ionization equilibrium is not main-tained for all times during the DM depletion phasecaused by bound-state decays. This is because, once thedecay rate exceeds the ionization rate, the system leavesthe ionization equilibrium. The temperature dependence

of the bound-state formation (BSF) and ionizationdetermines by how much the stable components aredepleted during this critical epoch. Once the BSFrate drops below the cosmic expansion rate H thedepletion stops.(iv) If we instead would have treated bound and

scattering states separately, with independent chemicalpotentials, we would have obtained Eq. (6.27). In thisequation one can see that the bound states are independentof the scattering states. Since the differential equation of theformer is linear in the number density on the r.h.s., at somepoint all bound states start to decay away. The couplingbetween bound and scattering states via radiative processesare not included in our theory and therefore do not appear.As we have learned, in our theory treating bound states ascomposite particles there is only one chemical potential.Therefore, by naively just giving bound and scatteringstates different chemical potentials would lead to the factthat we describe the system not with a grand canonicalensemble. Moreover, the KMS condition does not hold forthis case. In order to be able to introduce different chemicalpotentials might require to rewrite our theory in terms ofeffective operators, creating only scattering or only boundstates respectively. Then, there might be for every operatora conserved charge and one can associate individuallydifferent chemical potentials. We will see later that thismore “phenomenological” procedure is definitely notapplicable for finite temperature case. There it becomesimpossible to introduce such operators since the eigenval-ues might be not well defined when non-Hermitian thermalcorrections are included.(v) For going beyond the ionization equilibrium it is

required to include ultrasoft terms from the beginning andre-derive the DM correlator EoM including those correc-tions. This we leave for future work and we restrict ourequations to be valid as long as ionization equilibrium canbe maintained. If no bound-state solution exists, ourequations presented here only assume kinetic equilibrium[77,78]. If bound states exist, the validity of Eq. (6.26) canbe estimated. Once the decay rate exceeds the ionizationrate at late times, the regime of out-of-ionization equilib-rium starts (see also [21]). Until then, our description isvalid. In the next section, we will generalize this equationfor the finite temperature case. We will assume that thethermal width in the effective potential is small. Already inthis case, the number of bound states is dynamical since thescreening as well as the constant real part term in theeffective potential are temperature dependent. They lead fordecreasing temperature to an abrupt occurrence of bound-state poles in the spectral function (see also Sec. VII).Therefore, at finite temperature the description viaEq. (6.26) is insufficient. The more general descriptioncan be obtained from our formalism by going back to ourmaster equation, expressing the annihilation or decays interms of the two-particle spectral function. The spectral


115023-25

function automatically decides for a given temperaturewhich part of the spectrum contributes to the continuumand which part is bounded, as we will see later. Moreover,as we discuss in the next section, it is required at finitetemperature to include nonideal contributions to the ideal-ized Saha equation, leading to finite temperature correc-tions of the chemical potential. Thus, for a consistenttreatment we need to compute both: long-range modifiedannihilation and the chemical potential to the same order ofapproximation.

D. Chemical potential in narrow thermalwidth approximation

In the previous section, we established what is theoutcome of our master equation in the limit of vanishingthermal corrections. In this section, we compute thechemical potential for finite gχ and gψ . The onlyassumption we make is that the thermal width in themediator correlator D is subleading. Only the real-partcorrection in Eq. (5.32) is kept. This is called the narrowwidth (or quasiparticle) approximation. It leads to the factthat the correlator takes the simple form

Dðx; yÞ ¼ δCðtx; tyÞð−iÞVðx − yÞ; ð6:29Þ

where V is the screened Yukawa potential. To now evaluatethe pressure explicitly, we make use of the structure of theHamiltonian for this potential:

H ¼ H0 þMðNη þ NξÞ

þ g2χ2

Zx;y

Vðx − yÞ½η†ðxÞη†ðyÞηðyÞηðxÞ

þ ξðxÞξðyÞξ†ðyÞξ†ðxÞ − 2η†ðxÞξðyÞξ†ðyÞηðxÞ�:ð6:30Þ

Substituting g2χ → λg2χ and taking the partial derivative ofthe partition function with respect to λ we arrive at theconvenient form

pΩ − p0Ω ¼ TZ

1

0

dλ1

Zgr∂λZgrðΩ; T; μÞ ð6:31Þ

¼ TZ

1

0

dλZx;y

g2χ2Vðx − yÞ

× ½2Gþþ−−ηξ ðx; y;x; y; t − t0; λÞjt¼t0

− Gþþ−−ηη −Gþþ−−

ξξ �: ð6:32Þ

Spin indices are summed over equal arguments andGþþ−−ηη ,

Gþþ−−ξξ carry the same arguments as Gþþ−−

ηξ . Applying theKMS condition, and expressing Gþþ−− in terms of spectralfunction we arrive, after introducing Wigner coordinatesand Fourier transformation, at

βp ¼ βp0 þ e−β2MZ

d3Pð2πÞ3 e

−βP2=4M

ZdEð2πÞ e

−βEZ

1

0

dλ

×Zr

g2χ2VðrÞ

h2eβðμηþμξÞGρ

ηξðr; r;E; λÞ

− eβ2μηGρηη − eβ2μξGρ

ξξ

i: ð6:33Þ

This equation tells us that the total pressure is equal to theideal pressure, defined as

p0Ω ¼ T ln Tr½e−βðH0−μηNη−μξNξÞ�; ð6:34Þ

plus nonideal contributions arising from DM long-rangeself-interactions. It is possible to eliminate the λ integrationin Eq. (6.33) by partial integration and using the BSequation backwards. The final result can be expressed interms of bound-state contributions and the change of thescattering phase with respect to the energy for the scatteringstates. The equation is then known as Beth-Uhlenbeckformula. For the following discussion, it is however notrequired to explicitly give those expressions and thereforewe just refer to the result in standard text books for nonidealplasmas; see [81].10 Let us remark that one can also directlysolve Eq. (6.33) numerically via the solution of the two-particle spectral function. This is the power of Eq. (6.33).One can solve self-consistently for Gþþ−−

ηξ;s ðx; x; x; xÞjeq andthe chemical potential (see below) entering our masterequation just by evaluating the two-particle spectral func-tion without specifying what is a bound or a scattering state.The two-particle spectral function automatically takes intoaccount everything. This is because Eq. (6.33) is exact inthe narrow thermal width limit. We have just reduced theproblem of evaluating the total pressure to the evaluation ofa four-point correlation function.For the moment, only the structure of Eq. (6.33) is

important. Namely, when differentiating the total pressureequation with respect to the chemical potential we obtain,according to Eq. (6.3), for the total DM number densities

nη ¼ nη;0 þ KðTÞnη;0nξ;0; ð6:35Þ

nξ ¼ nξ;0 þ KðTÞnη;0nξ;0; ð6:36Þ

nη;0 ¼ neqη;0eβμη ; ð6:37Þ

nξ;0 ¼ neqξ;0eβμξ ; ð6:38Þ

10In nonideal plasma literature, the nonideal contribution inEq. (6.33) is referred to as the second-viral coefficient. It might bealso interesting to note that the number of bound states are relatedto scattering phases according to the Levinson theorem.


115023-26

neqη;0 ¼ neqξ;0 ¼ 2

�MT2π

�3=2

e−βM: ð6:39Þ

The subscript 0 labels ideal number densities, i.e., nη;0 is anumber density for a free DMwithout gauge interactions asin the previous section. Superscript “eq” stands for chemi-cal equilibrium and a symmetric plasma is assumedμη ¼ μξ. We stress that nη;0 should not be confused withnumber densities of one quasiparticle excitation of DM inthermal plasma. This equation just tells us that the exacttotal DM number density, nη and nξ, including correctionsfrom thermal plasma and bound states, can be simplyexpressed as Eqs. (6.35) and (6.36) by means of the idealnumber densities given in Eqs. (6.37)–(6.39). All theeffects from interactions are encoded in KðTÞ. An explicitexpression for KðTÞ can be obtained by comparingEq. (6.35) to (6.33). Again, it is only important to knowthat KðTÞ includes bound-state contributions as well asscattering parts which can be seen from the integration ofspectral function in the whole energy range. One can seethat in the limit of KðTÞ → 0 the number densities of freeDM are recovered. If we would send finite temperaturecorrections to zero and only include the bound-statecontribution, then KðTÞ just reproduces the ideal gas casefrom the previous section; see Eqs. (6.23) and (6.22). In thissense, standard Boltzmann equations (see also previoussection) typically used in numerical codes solve for the DMnumber density non-self-consistently. This is because theymiss the (small) nonideal corrections coming from thescattering contributions in KðTÞ and evaluate the chemicalpotential in the ideal gas approximation.For a symmetric plasma nη;0 ¼ nξ;0, Eq. (6.35) is a

quadratic equation in nη;0 and the solution is given by

nη;0nη

¼αðnηKðTÞÞ; αðxÞ¼ 1

2xð ffiffiffiffiffiffiffiffiffiffiffiffi

1þ4xp

−1Þ; ð6:40Þ

where α gives the ratio between the ideal number densityof free particles (no bound states and no interactions) andthe total number density nη, including bound states, non-ideal and thermal corrections. This is the main differencecompared to the ideal definition in the previous section.The generalized Saha Eq. (6.40) accounts for nonidealcontributions like self-interactions as well as for finitetemperature corrections. Important to note is that we do nothave to define what is a bound or a scattering contributionto the total number density. The spectral function inEq. (6.33) does the job automatically and takes all con-tributions into account when integrating over the wholeenergy range.Since nη;0 ¼ neqη;0e

βμη , we can finally determine thechemical potential μη as a function of total number densityfrom Eq. (6.40).Assuming a grand canonical state for a system having

bound-state solutions in the spectrum automatically implies

the Saha ionization equilibrium. Under this assumption, thechemical potential is set by

βμη ¼ ln

�αnηneqη;0

�; ð6:41Þ

and our master formula for the total number density Eq. (6.1)can be written in a fully closed form as

_nη þ 3Hnη ¼ −2ðσvrelÞGþþ−−

ηξ;s ðx; x; x; xÞjeqneqη;0ðTÞneqη;0ðTÞ

× ½α2ðnηKðTÞÞnηnη − neqη;0ðTÞneqη;0ðTÞ�;ð6:42Þ

where

Gþþ−−ηξ;s ðx; x; x; xÞjeq ¼ e−β2M

Z∞

−∞

d3Pð2πÞ3 e

−βP2=4M

×Z

∞

−∞

dEð2πÞ e

−βEGρηξð0; 0;EÞjl¼0:

ð6:43Þ

Importantly, the obtained number density Eq. (6.42) is ingeneral not quadratic in nη contrary to the conventional Lee-Weinberg equation because the generalized ionization frac-tion, αðnηKðTÞÞ, exhibits a nontrivial dependence on nη. Aswe have discussed in the previous section, this equationcontains all the number violating processes of DM, i.e.,annihilations and bound-state decay. The process whichdominates the decrease of DM number density is determinedby how α evolves in time as we have discussed at the end ofthe previous section.The insights we gained in this section are important for

the understanding of our work. Let us put below theseresults more into context.(i) Equation (6.42) describes the out-of-chemical equi-

librium (finite μ) evolution of the total number densityunder the constraint of ionization equilibrium and is one ofour main results. It is a generalization of the idealizedvacuum Eq. (6.26), accounting for (i) finite temperaturecorrections to the annihilation/decay rates and (ii) nonidealcorrections to the chemical potential. The nonideal correc-tions to the chemical potential consist of finite temperaturecorrections as well as scattering contributions. The mainadvantage of the Eq. (6.42) is that we do not have to definewhat is a bound or scattering state contribution since atfinite temperature this is meaningless to do. All expressionsneeded in order to solve our generalized number densityequation numerically can be obtained by evaluating thetwo-particle spectral function. Since one has to integrateover the whole energy spectrum, the result manifestly takesall contributions into account, without the need of differ-entiating between bound and scattering states. Later in


115023-27

Sec. VII, we show the results for the two-particle spectralfunction Gρ

ηξð0; 0;EÞjl¼0 in Eq. (6.42) including finitetemperature corrections.(ii) Let us discuss a bit more in detail the generalized

Saha Eq. (6.40). For vanishing finite temperature correc-tions and the ideal gas limit, we have emphasized that itreduces to the standard expression Eq. (6.24). Since (6.40)looks the same as the one in the ideal gas limit aside fromhow KðTÞ depends on T, one may understand how αevolves from the discussion at the end of the previoussection. However, at finite temperature it is more compli-cated to precisely estimate since the number of bound statesis dynamical. Furthermore, it is also more complicated todiscuss the case when ionization equilibrium is broken atfinite temperature due to this reason.Finite temperature effects might extend the period where

the ionization rate is much larger compared to the decayrate, since ψ particles can efficiently destroy the boundstate. Thus it might be true that the validity of our equationsholds longer compared to the vacuum case. Anotherdifficulty at finite temperature is, only in the limit ofnarrow thermal width it might be possible to estimate thevalidity of (6.42). This is because in order to estimate therates one has to define what is the decay width of the boundstate, which becomes hard to define beyond the narrowthermal width limit. At late times of DM freeze-out,however, we naively expect that the thermal width becomesless important and one may estimate the rates by justincluding the real-part corrections (but still in this case thenumber of highly excited bound states can be dynamical).(iii) According to previous discussions, we would like to

emphasize again that our number density Eq. (6.42) is notapplicable to the regime where bound-state decay ratesexceed the ionization rates at late times causing an out-of-ionization equilibrium state (now disregarding the issueof how we can precisely estimate those rates at finitetemperature). However, our more general equationsEqs. (4.20)–(4.21) do not assume ionization equilibriumand can be applied to any out-of-equilibrium state. Thepoint is, since we have dropped for simplicity soft emis-sions from the beginning, there are no processes like BSFvia the emission of a mediator relating the number of boundand free particles. Consequently, if one would solve thegeneral Eqs. (4.20)–(4.21) numerically, with an initial out-of-ionization equilibrium state, the system would remainfor all times in out-of-ionization equilibrium. It is requiredfor future work to include soft emissions via e.g., an electricdipole operator in thermal plasma, to account for a correctdescription of the DM thermal history at late times. We willsee, however, if ionization equilibrium can be guaranteed,our description accounts for sizable finite temperaturecorrections during the early phase of the freeze-outwhich can not be captured by the classical on-shellBoltzmann equation treatment as in [21]. Thus thesedifferent approaches are in some sense complementary.

Furthermore, the approach in [21] uses only the maincontribution of the ground state 1S, while via Eqs. (6.42)and (5.44) it is very elegant and efficient to include all (hereonly s-wave) bound-state contributions (but under theassumption of ionization equilibrium).(iv) From this section we learned that the standard

Boltzmann equations at zero temperature are a non-self-consistent set of equations. They miss scattering contribu-tions in KðTÞ which can now be fully accounted for.Bearing in mind that these contributions might be small, itmight also be acceptable to adopt a non-self-consistentsolution of Eq. (6.42). By this we mean one can in prin-ciple compute the chemical potential in the narrowwidth approximation; however, in the computation ofGþþ−−

ηξ;s ðx; x; x; xÞjeq, the two-particle spectral functioncan be solved in its most general form including finitetemperature width (as we present in Sec. VII). We empha-size again that a non-self-consistent solution might causesome troubles and care should be taken. In Appendix E 3,we discuss other non-self-consistent computations of thechemical potential and point out their failure, especially forlate times if bound-state solutions exist.

E. Comparison to linear-response-theory method

Our dark matter system is similar to heavy-quark pairannihilation in a thermal quark gluon plasma produced inheavy ion collisions. In literature, the annihilation rate of theheavy-quark pair into dileptons is estimated from linearresponse theory. Let us just quote their results in thefollowing without deeply diving into the details. Forcomparison we translate the expressions for SUð3Þ to ourU(1) by adjusting color and flavor factors and call the heavy(anti)quark dark matter (ξ) χ. The linearized Boltzmannequation around chemical equilibrium is given by

_nηþ3Hnη¼−Γchemðnη−nη;eqÞþOðnη−nη;eqÞ2; ð6:44Þ

where Γchem is called the chemical equilibration rate. Γchemcan be extracted from linear response theory, assumingthermal equilibrium and a perturbation linear around chemi-cal equilibrium. It is defined as [64–67]

Γchem ≡ Ωchem

ð4χη=βÞM2; ð6:45Þ

where Ωchem is a transport coefficient and χη is the heavyDM number susceptibility. The transport coefficient isquoted as

Ωchem=M2 ≃ 16ðσvrelÞZω;P

e−βð2MþωÞρηξðω;PÞ; ð6:46Þ

where we consider always only s-wave contributions and thetree-level annihilation cross section is defined as in our case,while ρηξ is also called a spectral function but might be


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defined slightly differently than ours. This expression lookssimilar to our term in chemical equilibrium

2ðσvrelÞGþþ−−ηξ;s ðx; x; x; xÞjeq

¼ 2ðσvrelÞZω;P

fBð2M þ ωÞGρηξð0; 0;ω;PÞ: ð6:47Þ

Indeed, we find both spectral functions in Eqs. (6.46)and (6.47) give identical results if we take the DM dilutelimit of our equation.The number susceptibility is defined as the response

of the total number density with respect to infinitesimalvariation of the chemical potential:

χη ≡ ∂nη∂μη

��μη¼0

: ð6:48Þ

In the dilute limit, using Eq. (6.35), we have χη ¼ βneqη .So we find in total for the linearised equation using linearresponse theory:

_nη þ 3Hnη ¼ −4ðσvrelÞ

Rω;P e

−βð2MþωÞρηξðω;PÞneqη n

eqη

× ½nηneqη − neqη neqη �: ð6:49Þ

In the following, we compare this equation based onlinear response theory with our Eq. (6.42). We start fromour more general expression and reproduce Eq. (6.49) inthe linear regime around chemical equilibrium whileclarifying the underlying approximations. For this purpose,we have to linearize our equation around nη ∼ neqη , where“eq” labels chemical equilibrium. On the one hand, aswe have shown in Sec. V, the spectral function, Gρ

ηξ, doesnot depend on nη as long as the DM number densities aredilute. On the other hand, the generalized ionizationfraction, α, does depend on nη and hence it must also beapproximated as to be close to chemical equilibrium. Onemay easily see this by looking at the definition of α given inEq. (6.40). From this expression it is clear that if the DMnumber densities are close to chemical equilibrium, theionization fraction is always close to one, i.e., αeq ≃ 1,because neqη;0 ∝ e−βM and KðTÞ ∝ eβjEBj. Hence, in thelinear regime, the total DM number densities are fullyionized and can be approximated as the free scatteringcontributions: neqη ≃ neqη;0. From these observations, wereproduce Eq. (6.49) from our Eq. (6.42) in the linearregime around chemical equilibrium.The linear response theory method applies, by definition,

only to the linear regime around chemical equilibriumwhere nη ∼ neqη . It is not possible via this method to seewhat is the correct form of the underlying Lee-Weinbergequation describing the nonlinear regime where nη ≫ neqη .Nevertheless, one might be tempted to use it by replacingthe right-hand side of Eq. (6.44) with Γchem

2neqηðn2η − ðneqη Þ2Þ as

done in [37,55–57,66]. Most important to note is that thecollision term obtained from this replacement matches ourequation only if the generalized ionization fraction is closeto chemical equilibrium, i.e., αeq ≃ 1, which is no longertrue at late times. We can see how α depends on time fromEq. (6.40). As we have already discussed in detail at the endof Secs. VI C and VI D, at late times of DM freeze-out thetotal comoving number density approaches a constant valuewhile KðTÞ starts to grow rapidly for T < jEBj. As a result,we find the generalized ionization fraction to be α ≃1=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinηKðTÞp

≪ 1 at late times, which invalidates the naivereplacement above. One can understand this behaviorintuitively because the bound states of energy 2M − jEBjare thermodynamically favored compared to the scatteringstates of energy 2M at T < jEBj. For a fixed total number ofDM, the bound states dominate over the scattering states atsome point as in the case of the recombination. There isalso another issue when correcting neqη in Γchem

2neqηðn2η − ðneqη Þ2Þ

only by the Salpeter term. Since this discussion requiressome detailed knowledge about thermal corrections, weshare it in Appendix E 3. In summary, we would like toemphasize that care must be taken if one matches theequation obtained by linear response theory to nonlineardifferential equations.Instead, one may match the equation obtained by linear

response theory to our corrected form of the Lee-Weinbergequation in the nonlinear regime. By this procedure it is nowalso clear what the limitation exactly is. Aswe have discussedin detail in the previous section, our master formula for thenumber density equation is valid as long as ionizationequilibrium can be maintained. Ionization equilibrium isbroken if the decay rate exceeds the ionization rate. Thetemperature where this happens can be estimated for finitetemperature systems, at least in the narrow width case.

VII. NUMERICAL RESULTS FOR TWO-PARTICLESPECTRAL FUNCTION AT FINITE

TEMPERATURE

We turn to the numerical solution of the two-particlespectral function for the full in-medium potential. Theeffects of the finite temperature corrections can be simplestunderstood for the case of the Coulomb potential as givenin Eq. (5.32). The first correction is a real constant termthat shifts effectively only the energy by αχmD. When onlytaking this correction into account one would thus expectthat the infinite number of bound states in the spectralfunction of the Coulomb case just move to lower bindingenergies and similar shift to the threshold as well as to thepositive energy spectrum. The second real-part correction isan exponential screening of the Coulomb potential withradius mD. This introduces another effect. It leads to adisappearance of the bound states closest to the thresholdsince Yukawa potentials have only a finite number of boundstates. The disappearance of bound states wins against


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the move of the poles towards lower energies at the Motttransition, where all bound states disappear and thespectrum is exclusively continuous. Additionally, we haveimaginary-part corrections coming from the soft DM-ψscatterings, leading to a finite thermal width of the boundstates. Once the thermal width is comparable to the bindingenergy, the bound-state poles are strongly broadened.The combination of all effects are shown in Fig. 4, where

wepresent the numerical solutionof the two-particle spectralfunction for the full in-medium potential according toEq. (5.44). We show the case of the Coulomb andYukawa potential. In this figure, we have fixed the temper-ature toT ¼ M=30 (slightlybelow the typicalDMfreeze-outtemperature) and varied the Debye mass where the maximalvalue shown corresponds to the equal charge case gψ ¼ gχ ofour minimal model: m2

D ¼ g2χT2=3. The mass of the DM isfixed to 5 TeV and the coupling αχ ¼ 0.1 very roughlychosen to account for the correct order of the abundance.All finite temperature effects together lead to a continu-

ous melting of the bound-state poles. As can be seen, themelting of the bound states leads to the fact that even atnegative energies, the spectrum is continuous at finitetemperature. The reshuffling of the spectrum towards lowerenergies affects the rates exponentially according toEq. (5.8). This is because the integrand (the spectralfunction) has more support at negative energies whichis, due to Boltzmann factor, exponentially preferred inkinetic equilibrium. It now becomes clear that the notion ofspectral function is more general compared to the vacuumcase where one could separate the spectrum for bound andscattering states. Here, it is evident that such a distinction isimpossible. It is also not necessary to do so since theintegration of spectral function times Boltzmann factortakes already all contributions into account.In all cases we study in this section, the integral has a fast

convergence at negative energies well inside the validityregion of HTL resummed effective theory jEj≲ T. An

intuitive reason why our treatment breaks down for a largenegative energy −E ≫ T is the following. In this regimedark matter and antidark matter are tightly coupled. As aresult, typical scatterings with momentum exchange of Tcannot probe inside the dark matter and antidark matterpair. Thus, we expect the imaginary part in the effectivepotential for jEj ≫ T to be suppressed by an additionalBoltzmann factor. In a word, for large enough jEj, thesituation should revert to the vacuum case and we no longerexpect thermal corrections for the bound states. Indeed, forvanishing imaginary part in the potential the two-particlespectral function has no support for negative energiesbelow the ground state energy.Nevertheless, let us discuss how only the positive energy

spectrum is affected. According to the theorem ofLevinson, the scattering phases (and hence the wavefunction at the origin) depend on the amount or propertiesof the bound states. This means that thermal modificationsof the bound states automatically affect also the positiveenergy spectrum. The impact on the positive energyspectrum depends on the melting status of the boundstates. In general, there can be both, a suppression orfurther enhancement of the positive energy spectrum as canbe seen by carefully looking at the value around E ¼ 0 inFig. 4. We would like to stress that a suppression of thepositive spectrum does not imply that the total rate is less.For the computation of the rate one has to integrate thespectral function over the whole energy range, whereGþþ−−

ηξ;s ðx; x; x; xÞjeq becomes due to the reshuffling towardslower energies exponentially enhanced.While for the Coulomb case, the impact of the melting on

the positive energy spectrum is only very little (which doesnot mean that the overall effect is small), the impact forthe Yukawa potential case can be much larger. In Fig. 5, wecompare the positive energy solution of the Yukawa spec-trum at zero and finite temperature, as a function of themediator massmV . The vacuum line (dashed) is obtained by

FIG. 4. Two-particle spectral function at finite temperature shown vs the energy in units of typical freeze-out temperature. The violetline corresponds to the equal charge case gψ ¼ gχ of our minimal model and hence m2

D ¼ g2χT2=3.


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solving the spectral function in the limit of vanishing finitetemperature corrections. The Sommerfeld factor for this casecan be obtained from Eq. (6.12) according to

SðvrelÞ ¼2π

M2vrelGρ

ηξð0; 0;EÞjE>0;l¼0 ð7:1Þ

⇔ SðEÞ ¼ π

MffiffiffiffiffiffiffiffiME

p Gρηξð0; 0;EÞjE>0;l¼0: ð7:2Þ

In the second line we used E ¼ Mv2rel=4 for on-shellparticles. At finite temperature this kinetic energy relationdoes not hold. We therefore use the second Eq. (7.2) todefine the Sommerfeld enhancement factor at finite temper-ature as shown in Fig. 5. The enhancement or suppression ofthe Sommerfeld enhancement factor due to the thermaleffects is largest if the ground state is close to the threshold ofE ¼ 0 (around the first peak from the right, here ϵϕ ∼ 0.6).For our minimal model, it is also shown in the right plot of

Fig. 5 that the whole temperature range of a typical freeze-out process can be affected. In the limit mV → 0 theCoulomb limit is recovered. Again, this does not mean thatit is sufficient to just take the standard expression of theSommerfeld enhancement factor of the Coulomb potential todescribe the DM freeze-out. There is also a contribution fromthe negative energy spectrum. Therefore, one has to becareful in interpreting Fig. 5. On the one hand side thepositive energy solution (Sommerfeld enhancement factor)can be suppressed or equal compared to the vacuum case, buton the other side the total Gþþ−−

ηξ;s ðx; x; x; xÞjeq entering ourmaster formula Eq. (6.1), which requires the integration overthe whole energy spectrum, can be enhanced.As an extreme example of this situation, let us discuss

the case where there are no bound states (e.g., Yukawapotential in the Born regime ϵϕ ≫ 1). The finite temper-ature spectral function for this case is shown in the rightplot of Fig. 6. Indeed, the positive energy spectrum is

FIG. 5. Sommerfeld enhancement factor for a Yukawa potential shown vs the mediator mass in units of the Bohr radiusϵϕ ≡mV=ðαχMÞ. Dashed and solid lines correspond to the vacuum limit and the full in-medium potential, respectively. Effects atdifferent temperatures are compared. Left side for typical freeze-out temperature and right plot at a typical temperature where theannihilation rate would be much smaller compared to the Hubble rate.

FIG. 6. Comparison between extreme examples, where only the ground state exists and is close to the threshold (left) and where nobound states exist (right). Here, the vacuum curve is defined as not including delta-peaked bound-state contributions.


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dominantly suppressed compared to the vacuum case butwhen integrating the spectral function over the Boltzmannfactor in the whole energy range there is still an enhance-ment of order 1% compared to the vacuum case. Anotherextreme example, where the corrections to the positivespectrum are strongest, is the case where only the groundstate exists and is close to threshold [37,55,56]. Thisexample is shown in the left plot of Fig. 6. We find inthis case, the value of the integration of the spectralfunction times the Boltzmann factor is by up to 10%(30%) larger compared to the vacuum case without bound-state peak at the typical freeze-out temperature T ¼ M=30(T ¼ M=90). The correction increases for lower temper-ature due to the Boltzmann factor.The total rate is only proportional to the integration of the

spectral function times the Boltzmann factor. There areadditional finite temperature corrections to the chemicalpotential (see previous section) which can be also obtainedfrom the spectral function. We have not explicitly com-puted those nonideal corrections yet but leave it for futurework once we have included ultrasoft emissions in oursystem description.

VIII. DISCUSSION

A self-interacting DM system, where long-range forcesand bound-state solutions exist, is in general a complexensemble where many processes with different rates aretaking place at the same time during the DM thermalhistory. Essentially, there are three quite differentapproaches in the literature with distinct motivation todescribe the evolution of the abundance of the stablecomponents for such systems:(1) The first approach is based on a coupled set of

classical on-shell Boltzmann equations. If bound-statesolutions are absent, the description of the DM freeze-out acquires dominantly corrections from the Sommerfeld-enhanced annihilation of free DM particles [10,11]. If thetwo-particle spectrum has support at negative energies, thefree DM particles can form a bound state via radiativeprocesses [21,82]. The reverse process can also happen,called ionization. If there are several bound-state solutionspresent, further processes like excitation or deexcitation canhappen [22,42]. All those processes are in general coupled,and as we see, the list of Boltzmann equations needed todescribe such systems can be long. When relying on thoseclassical Boltzmann equation computations, treating e.g.,the number density of free particles and bound statesseparately and as idealized, potential strong modificationsarising from higher-order plasma interactions might bemissed. In this approach, however, it is always guaranteedthat the nonlinearity of out-of-chemical equilibrium reac-tions are accurately described. And there can be in generalmany such out-of-equilibrium reactions as listed above.(2) The second approach starts from the EoM of

correlation functions on the Keldysh contour and takes

into account some finite temperature corrections. The majordifference to our work is that in [69] it is assumed that thecorrelator hierarchy can by truncated at the lowest order,resulting in closed equations for the two-point functions interms of the one-particle self-energy only. One of theequations are the so called kinetic equations, being thedifferential equations for describing the evolution ofobservables in terms of the macroscopic Wigner coordi-nates. In the one-particle self-energy approximation theyare also known as Kadanoff-Baym equations. Expandingthe self-energy in terms of the coupling to NLO results inthe standard Boltzmann equation. At NNLO first finitetemperature corrections enter. The advantage of a fixedorder calculation is that infrared divergences, arising atNNLO cancel [69]. At NNLO in the self-energy expansionof the kinetic equations, the thermal corrections turn out tobe strongly suppressed, i.e., to high power in T=M,compared to the NLO result. One should, however, keepin mind that there are next to the kinetic equations also theequations for the microscopic Wigner coordinates, calledmass-shell equations accounting for, e.g., thermal correc-tions to the dispersion relation. Kinetic and mass-shellequations are in general coupled. Therefore, a self-con-sistent solution in principle requires to take account ofcorrections also from the mass-shell equation. In any case,the problem within this systematic approach is that a fixedorder calculation can never account for correctly describingthe Sommerfeld enhancement beyond the Born regime andalso bound-state solutions will never appear.(3) The third approach addresses the description of long-

range force systems at finite temperature in a nonperturba-tive sense, i.e., by resummation of the Coulomb divergentladder diagrams including thermal corrections. Clearly, firstattempts were made in the literature of heavy quark pairannihilation in a quark gluon plasma [64], produced inheavy-ion collisions at the LHC. More recently, some ofthese authors have applied the same techniques also tothe DM freeze-out [37]. The method is based on linearresponse theory [65–67], estimating the DM Sommerfeld-enhanced annihilation and bound-state decay from aspectral function including finite temperature corrections.It has been shown in Ref. [55–57] that the DM overclosurebound, computed by this method, can be strongly affectedby finite temperature effects if bound-state solutions exist.Compared to a fixed order calculation as in approach 2, thefinite temperature corrections are larger. The reason isbecause the mass-shell equations are solved by resumma-tion of the Hard thermal loop contribution. Albeit there arepotentially strong effects, the linear response theory isstrictly speaking valid only for systems close to thermalequilibrium, e.g., n ∼ neq. At finite temperature the spectralfunction can in general depend on the DM density.Therefore, it is a priory not clear if the transport coefficientsextracted from linear response theory can be insertedinto a nonlinear Boltzmann equation describing the DM


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freeze-out in a nonlinear regime where n ≫ neq. Fromvacuum computations it is known that the Sommerfeldeffect can still be efficient in such a regime. This is becausethe transition n ∼ neq to n ≫ neq happens in a short time,since neq ∝ e−M=T decreases rapidly. To the best of ourknowledge this method, inserting transport coefficientsobtained from linear response theory into a nonlinearBoltzmann equation, has not been tested so far by usingother treatments applying for generic out-of equilibriumsituations.Our formalism, presented in this work, aims towards a

first step in unifying the approach 1 and 3, by generalizingthe approach 2 for long-range force systems. In otherwords, we derived from the EoM of Keldysh correlationfunctions the number density equation for DM includingfinite temperature corrections and accounting for thefull resummation of Coulomb divergent ladder diagrams.This allows to study the finite temperature correctedSommerfeld-enhanced annihilation as well as bound-statedecay. Moreover, our master Eq. (6.1) is able to describe thecorrect nonlinear transition to out-of-chemical equilibrium,i.e., the freeze-out process.Although we have derived all equations on the Keldysh

contour, and therefore they should be valid for any out-of-equilibrium situation of the system, the reader should bereminded what precisely our system is. While it remainstrue that we can describe correctly Sommerfeld-enhancedannihilation and bound-state decay in the presence of arelativistic plasma background for out-of equilibrium sit-uations, we have dropped from the beginning, whenderiving our nonrelativistic effective action, ultrasoftcontributions of the fully relativistic action. Hence,Eqs. (4.20)–(4.21) are missing ultrasoft contributionsleading to bound-state formation and ionization processesvia the emission or absorption of a mediator, as well ascontributions to excitation or de-excitation processes ifmultiple bound states exist. Once ultrasoft terms areincluded in the system of equations, we expect the finalequations, if finite temperature effects are neglected, tocoincide with the full set of equations of approach 1.Moreover, the inclusion of emission and absorption in theKeldysh formalism might lead to new insights in theproduction rate of dileptons or photons, produced fromheavy-quark pair annihilation in a quark gluon plasma.In the second half of the work, we have indirectly

included all bound-state formation, ionization, excitation,and deexcitation processes. This was achieved by assumingour system is in a grand canonical state with one single timedependent chemical potential as in our master Eq. (6.1).The important observation that adopting a grand canonicalpicture automatically implies ionization equilibrium ifbound states are present was by far not obvious to us.This key observation brought us to the conclusion that ourequations in the limit of vanishing thermal corrections areequivalent to the coupled system of classical Boltzmannequations in the limit of ionization equilibrium. Thus we

have shown that under certain assumptions our approachand approach 1 consistently fall together. Another impor-tant point based on this observation was that, since theionization fraction at chemical equilibrium is close to unity,our and approach 3 are equivalent to approach 1 in theregime linear near chemical equilibrium.Important to recognize was that our approach and

approach 3 give different results if the transport coefficientsextracted from linear response theory is inserted into anonlinear Boltzmann equation just by replacing Γchemðnη −neqη Þ with Γchem

2neqηðn2η − ðneqη Þ2Þ [55–57]. This is because the

ionization fraction depends on nη where another non-linearity comes in, and in particular, the ionization fractionwill be much smaller than unity at late times. This isintuitively because the bound states are exponentiallyfavored compared to the scattering states for T < jEBj.Furthermore, the ionization fraction counteracts against theexponential grow of Γchem or of our Gþþ−−

ηξ;s ðx; x; x; xÞjeq forlate limes if bound-state solutions exist. In a word, whilethe spectral function is identical between the linearresponse and ours in the DM dilute limit, the ionizationfraction makes the difference. This effect is non-negligiblewhen at late times the DM gets depleted by bound-stateformation effects.Our master Eq. (6.1) cannot be used at very late times

where ionization equilibrium is not maintained. Therefore,one has to be careful in relying on our so far simplifiedtreatment for all times during the DM thermal history. Moregenerally, when using our equations, it has to be ensuredthat the rates driving the system to kinetic and ionizationequilibrium are much faster than any other rates leading to apotential out-of-kinetic or -ionization equilibrium state. Inthe case of no ψ particles (no finite temperature correc-tions), it was shown in Refs. [21,22,42] that the decay of thebound state becomes faster than the ionization via emissionand absorption processes by an electric dipole operatorat some point, which breaks the ionization equilibrium atlate times. Later, when the bound-state formation becomesinefficient compared to the cosmic expansion, the darkmatter number freezes out completely. Estimating the validregime of our approach in the presence of finite temperaturecorrections is a more complicated task. To draw a definitiveconclusion in our case, one has to estimate these processes,including emission and absorption of ultrasoft gaugebosons, in the presence of the thermal plasma. As wehave discussed already in detail, this might only berealizable if the thermal width is negligible compared tothe real-part corrections. Furthermore, one has to keep inmind that the number of existing bound-state solutions istemperature dependent when already only real-part correc-tions are taken into account.After this warning, we now would like to discuss the

case where a grand canonical description with one singlechemical potential is justified. As we have in detailpresented in this work, all finite temperature corrections


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are then totally encoded in the solution of the DM two-particle spectral function. In the presence of ultrarelativisticfermionic particles in the background, the hard thermalloop resummed corrections to the DM system can beclassified into three contributions [see effective in-mediumpotential Eq. (5.32)]. The first two contributions are real-part corrections to the DM effective in-medium potential.The first one leads to an energy shift (Salpeter correction) inthe DM two-particle spectral function by an amount ofαχmD towards lower energies compared to the vacuumcase. Second, the Debye mass mD leads to a screening ofthe Coulomb potential, resulting in a temperature depen-dent Yukawa potential with screening radius mD. Thethird contribution to the effective potential is purelyimaginary and originates from soft DM scattering proc-esses with the ultrarelativistic fermions in the hot anddense background.Let us first discuss the two real-part corrections and their

implication on the two-particle spectral function for thecase of a Coulomb potential. In the limit of vanishing real-part corrections, it is well known that a Coulomb systemhas infinitely many bound-state solutions. Furthermore, thebound-state solutions and the scattering states can clearlybe separated sharply at the energy E ¼ 0 in the two-particlespectral function (see Fig. 3). If the real-part finite-temperature corrections are included, bound states closeto the threshold E ¼ 0 disappear in the spectrum. This issimple to understand. First of all, by the real-part correc-tions the Coulomb potential transforms into a Yukawapotential. Yukawa potentials have only a finite number ofbound states. Secondly, due to the energy shift caused bythe other real-part correction, the threshold is lowered. Thecombination of these two effects causes the disappearanceof highly excited bound states close to the threshold and thespectrum is continuous instead of discrete. The effect getsstronger with increasing Debye mass leading at the Motttransition to a total disappearance of all bound states.Already when only real-part corrections to the in-

medium potential are included, the number of boundstates as well as their binding energies are temperaturedependent according to the discussion above. It impliesthat a sharp definition of bound and scattering states can notbe made for all times during the DM thermal history.However, for our total number density equation, it is NOTrequired at all to distinguish between bound and scatteringstates. For example, the computation of the ionizationfraction via the generalized Saha Eq. (6.40) can always beperformed without specifying what is a bound or scatteringcontribution. Another example is the spectral functionentering the total number density in the production term.Also here, the integration of the spectral function auto-matically takes into account all contributions from thespectrum. Only in the absolute vacuum limit, it is possibleto separate contributions. At finite temperature everythingis mixed into one single object. A separation would only

cause problems like unphysical jumps in projected ther-modynamical quantities when bound states abruptly dis-appear (if only real-part corrections are included).The mixing between scattering and bound-state solutions

becomes even stronger once the imaginary contributions tothe effective in-medium potential are included. As we haveseen in Sec. VII, these corrections lead to a thermal widthof the peaks of the bound states and a continuous melting ofthe poles for increasing Debye mass, as illustrated in Fig. 4.Note that instead of changing the independent coupling inthe Debye mass, we could have also increased the numberof generations of ultrarelativistic particles in the plasma.The Debye mass is proportional to the square root ofnumber of generations, and thus, if most of the particlecontent of the Standard Model would run in the thermalloop, we can have a large Debye mass although thecoupling is still small. The broadening of the peak andthe shift towards lower binding energies increases theannihilation or decay rate exponentially (again its notnecessary to distinguish between these two) due to theintegration of the product of two-particle spectral functionand the Boltzmann factor, as in Eq. (5.8).Although, we focused on a simple U(1) like theory and

s-wave contributions in the present work, most of theequations for higher gauge theories, scalar mediators,or higher partial waves will change in the expected way.Let us already mention some major changes. The mediatorself-energy would acquire further contributions fromself-interactions as well as different colour or flavorprefactors. The definition of the effective in-mediumpotential in Eq. (5.31) remains the same and is computedfrom the specific dressed mediator correlator. The r.h.s. ofthe Schrödinger-like Eq. (5.30) will be proportional to thenumber of colours. Our number density Eq. (3.24) in thecase of velocity dependent tree-level annihilation crosssections (like in p-wave case) will have a space derivativeon the r.h.s. As expected, our formalism breaks down fortemperatures around the confining scale of confiningtheories.

IX. SUMMARY AND CONCLUSION

Traditional computations of DM Sommerfeld-enhancedannihilation and bound-state decay rates rely on theassumption that reactions of such processes are takingplace under perfect vacuum conditions. In this work wedeveloped a comprehensive derivation of a more generaldescription, taking into account nonideal contributionsarising from simultaneous interactions with the hot anddense plasma environment in the early Universe. We havederived the evolution equation for the DM number densitywhich is applicable to the case where scattering and boundstates get strongly mixed due to the influence of the thermalplasma surrounding. Our master Eq. (6.1) for the totalDM density simultaneously accounts for annihilation andbound-state decay and hence its collision term is in general


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not quadratic in the DM number density. We showed thatfinite temperature effects can lead to strong modificationsof the shape of the two-particle spectrum, which in turnmodifies the DM annihilation or decay rates.The Keldysh formalism we adopted throughout this

work applies for the description of the dynamics of genericout-of-equilibrium states. Within this mathematical frame-work, we derived in the first part of this work directly fromour nonrelativistic effective action the exact equationof motion of the DM two-point correlation functions.We extracted for the first time from those EoM thedifferential equation for the DM number density [seeEqs. (3.24) and (3.25)], which turns out to only dependon a special component of the DM four-point function onthe Keldysh contour, namely Gþþ−−

ηξ . Let us emphasizeagain that this equation for the number density is exactwithin our nonrelativistic effective action; however, it isnot closed since it depends on the solution of this four-pointcorrelation function. The long-range force enhancedannihilations, the decay of bounded particles as well asthe finite temperature corrections are all contained in thesolution of this one single four-point correlator.In the second part of this work, we derived the EoM

for the DM four-point function on the Keldysh contour.We developed the approximations needed in order to closethe hierarchy of correlators but at the same time keep theresummation of Coulomb divergent ladder diagrams aswell as the finite temperature corrections. Based on ourapproximation and resummation scheme, the final form ofthe equation for our target component Gþþ−−

ηξ is physicallysound and maintains important relations like the KMScondition in equilibrium. The coupled system of equationsis general enough to apply for the description of DM out-of-chemical equilibrium states.In the third part, we explored further approximations

needed in order to obtain a simple solution to our targetcomponent and to reproduce from our general equations theresults in the literature, based on different assumptions. Sofar existing literature has estimated transport coefficientsfrom linear response theory and entered those into anonlinear Boltzmann equation by classical rate arguments[37,55–57,64–67]. We have proven that our masterEq. (6.1) is equivalent to the method of linear responseonly in the linear regime close to chemical equilibrium.Finally, we must point out that the Lee-Weinberg equation,adopted in [37,55–57,66] to rederive the DM overclosurebound in the nonlinear regime, is not the correct form of thenumber density equation to use if bound-state solutionsexist in the spectrum. The ionization fraction causes thedifference as discussed in great detail in our work.When taking the vacuum limit, our master equation

reduces correctly to the coupled system of classicalBoltzmann equations for ideal number densities of boundand scattering states in the limit of ionization equilibrium.In our method, it came out as a consequence of assuming

the system is in a grand canonical state. Namely, we haveproven that the assumption of a grand canonical stateautomatically implies the Saha ionization equilibrium ifbound-state solution exist. One has to take the assumptionof ionization equilibrium to be fulfilled for all times with agrain of salt for the following reason. From the vacuumtreatment it is known that the duration of Saha ionizationequilibrium is limited. Therefore, when using our masterequation one has to carefully check that this condition issatisfied for a sufficiently long period. And especially whenthe assumption of ionization equilibrium is not justified,one has to make sure that at least the abundance of thestable scattering states are not affected by out-of-ionizationequilibrium effects which might be model dependent.The reason why in our Keldysh formalism we cannot

resolve this issue at the moment lies in one particularapproximation, made from the beginning. Ultrasoft emis-sions and absorptions were dropped for simplicity. Weleave the inclusion of those quantities for future work, butexpect once they are included we can fully recover thegeneral set of coupled classical Boltzmann equations in thevacuum limit of our (future) updated equations. Moreover,this would allow us to describe Sommerfeld-enhancedannihilation and bound-state decay at finite temperaturefor the first time beyond the ionization equilibrium.In the regime where ionization equilibrium is main-

tained, we have shown that finite temperature effectsstrongly mix bound and scattering states and the effectsare all encoded in the solution of the two-particle spectralfunction. Let us remark that the numerical results for thespectral function obtained in Sec. VII are compatible withthe linear response theory approach [37,55–57,64–67],although we started from a completely different method.The component Gþþ−−

ηξ;s jeq in our master Eq. (6.1) can beenhanced by much more than 10%. In addition, our masterequation is applicable to the nonlinear regime beyond thelimitation of linear response if, at least, the ionizationequilibrium is maintained. These results make it definitelyworthwhile to further generalize our Keldysh description inorder to correctly describe the out-of-ionization equilibriumtransition at late times by including contributions from theultrasoft scale.

ACKNOWLEDGMENTS

We are thankful to Kalliopi Petraki, Andrzej Hryczuk,Mikko Laine, and Shigeki Matsumoto for very usefulcomments on our manuscript. T. B. thanks the DESYTheory Group and Kavli IPMU for hospitality during anearly phase of this project. This work is supported byGrant-in-Aid for Scientific Research from the Ministry ofEducation, Science, Sports, and Culture (MEXT), Japan,World Premier International Research Center Initiative(WPI Initiative), MEXT, Japan, and the JSPS ResearchFellowships for Young Scientists (K. M.). T. B. and L. C.received funding from the German Research Foundation


115023-35

[Deutsche Forschungsgemeinschaft (DFG)] through theInstitutional Strategy of the University of Göttingen(RTG 1493), the European Union’s Horizon 2020 researchand innovation program InvisiblesPlus RISE under theMarie Sklodowska-Curie Grant Agreement No. 690575,and from the European Union’s Horizon 2020 research andinnovation program Elusives ITN under the MarieSklodowska-Curie Grant Agreement No. 674896.

APPENDIX A: SEMIGROUP PROPERTY OFFREE CORRELATORS

Free correlators G0 fulfil semigroup properties:

GR0 ðx; yÞ ¼ þ

Zd3zGR

0 ðx; zÞGR0 ðz; yÞ; for tx > tz > ty;

ðA1Þ

GR0 ðx; yÞ ¼ þ

Zd3zd3wGR

0 ðx; wÞGR0 ðw; zÞGR

0 ðz; yÞ;

for tx > tw > tz > ty; ðA2Þ

GA0 ðx; yÞ ¼ −

Zd3zGA

0 ðx; zÞGA0 ðz; yÞ; for tx < tz < ty;

ðA3Þ

GA0 ðx; yÞ ¼ þ

Zd3zd3wGA

0 ðx; wÞGA0 ðw; zÞGA

0 ðz; yÞ;

for tx < tw < tz < ty; ðA4Þ

Gþ−=−þ0 ðx; yÞ ¼ þ

Zd3zGR

0 ðx; zÞGþ−=−þ0 ðz; yÞ;

for tx > tz; ðA5Þ

Gþ−=−þ0 ðx; yÞ ¼ −

Zd3zGþ−=−þ

0 ðx; zÞGA0 ðz; yÞ;

for tz < ty; ðA6Þ

Gþ−=−þ0 ðx;yÞ¼−

Zd3zd3wGR

0 ðx;wÞGþ−=−þ0 ðw;zÞGA

0 ðz;yÞ;

for tx>tw and tz<ty: ðA7Þ

Note, there is no time integration here. All relations followfrom the first Eq. (A1) by using Eq. (2.13). It might behelpful to prove the first equation from definition, wherewe have

GR0 ðx; yÞ ¼ θðtx − tyÞ½G−þ

0 ðx; yÞ −Gþ−0 ðx; yÞ� ¼ Gρ

0ðx; yÞ;for tx > ty: ðA8Þ

The free spectral function for nonrelativistic particles is in

Fourier space given by Gρ0ðω;pÞ ¼ ð2πÞδðω − p2

2mÞ. Then itfollows:

Zd3zGR

0 ðx; zÞGR0 ðz; yÞ ðA9Þ

¼Z

d3zGρ0ðx; zÞGρ

0ðz; yÞ; for tx > tz > ty ðA10Þ

¼Z

d3zZ

d3pð2πÞ3

dωð2πÞ ð2πÞδ

�ω −

p2

2m

�

× eiðωðtx−tzÞ−p·ðx−zÞÞZ

d3p0

ð2πÞ3dω0

ð2πÞ ð2πÞ

× δ

�ω0 −

ðp0Þ22m

�eiðω0ðtz−tyÞ−p0·ðz−yÞÞ ðA11Þ

¼Z

d3pð2πÞ3 e

iðp22mðtx−tyÞ−p·ðx−yÞÞ ðA12Þ

¼ GR0 ðx; yÞ; for tx > ty:□ ðA13Þ

APPENDIX B: ANNIHILATION TERM

Although one can directly compute Γsðx; yÞ defined onthe closed-time-path contour, it is instructive to see howone can recover it from the more common computation ofannihilations. Usually, we compute the matrix elementby means of Feynman correlators so as to evaluateannihilations, which means that both x and y are on theCþ contour. And thus, it yields the upper left component ofΓs, i.e., Γþþ

s . The question is how to recover the remainingthree components. To answer it, let us go back one stepfurther, namely before integrating out hard products of theannihilation. Suppose that the interaction with them takesthe following form: OsðxÞOH½χðxÞ�e−2iMx0 þ H:c:, with χbeing the products of the annihilation. HereOs is defined asEq. (3.4) andOH represents hard d.o.f. Then, Γs is obtainedfrom the cuttings of hTCO

†H½χðxÞ�OH½χðyÞ�ie2iMðx0−y0Þ.

Since we assume that the background plasma is in thermalequilibrium and does not change by η and ξ reactions, thetwo-point correlator of OH only depends on the space-timedifference x − y:

GOHðx − yÞ≡ hTCO

†H½χðxÞ�OH½χðyÞ�i: ðB1Þ

By definition, incoming energy/momentum from Osis much smaller than M, which justifies the followingapproximation:Z

x;y∈CþO†

sðxÞOsðyÞiGþþOH

ðx − yÞe2iMðx0−y0Þ

≃Zx;y∈Cþ

O†sðxÞOsðyÞiGþþ

OHð2M; 0Þδðx − yÞ: ðB2Þ

Taking the imaginary part and comparing it with Γþþs ,

one can see that the usual computation corresponds toℑiGþþ

OHð2M; 0Þ. As the background plasma is assumed to

be close to equilibrium, we can use the Kubo-Martin-


115023-36

Schwinger (KMS) relation, which essentially connects allthe other combinations, G−þ

OH, Gþ−

OH, G−−

OH, with this one.

Moreover, because ofM ≫ T, one can safely neglect Gþ−OH

.As a result, we end up with

ℑiGþþOH

ð2M; 0Þ ¼ πðα2χ þ αχαψ ÞM2

; ðB3Þ

ℑiGþ−OH

ð2M; 0Þ ¼ 0; ðB4Þ

ℑiG−þOH

ð2M; 0Þ ¼ ℑiGretOH

ð2M; 0Þ − ℑiGadvOH

ð2M; 0Þ¼ 2ℑiGþþ

OHð2M; 0Þ; ðB5Þ

ℑiG−−OH

ð2M; 0Þ ¼ −ℑiGretOH

ð2M; 0Þ þ ℑiG−þOH

ð2M; 0Þ¼ ℑiGþþ

OHð2M; 0Þ; ðB6Þ

which results in Eq. (3.4). Here, several properties ofequilibrium correlators were used, as given in Sec. II.

APPENDIX C: HARD THERMAL LOOPAPPROXIMATION

The Fourier transform of the ideal ψ two-point functionis in the Keldysh representation given by

SðPÞ ¼ ðPþmÞ�� i

P2−m2þiϵ 0

0 −iP2−m2−iϵ

�− 2πδðP2 −m2Þ

×

�nFðjp0jÞ −θð−p0Þ þ nFðjp0jÞ

−θðp0Þ þ nFðjp0jÞ nFðjp0jÞ

��:

ðC1Þ

Combining different components, the following retarded,advanced, and symmetric propagator can be obtained:

SR=AðPÞ ¼ iðPþmÞP2 −m2 � isignðp0Þϵ ; ðC2Þ

SsðPÞ≡ SþþðPÞ þ S−−ðPÞ¼ 2πðPþmÞ½1 − 2nFðjp0jÞ�δðP2 −m2Þ: ðC3Þ

The inverse relations are given by

Sþþ ¼ 1

2ðSs þ SR þ SAÞ;

Sþ− ¼ 1

2ðSs − SR þ SAÞ;

S−þ ¼ 1

2ðSs þ SR − SAÞ: ðC4Þ

By using these relations, the one-loop expression of theretarded mediator correlator can be simplified as

Tr½γμSþþðx;yÞγνSþþðy;xÞ�−Tr½γμSþ−ðx;yÞγνS−þðy;xÞ�

¼1

2Tr½γμSRðx;yÞγνSSðy;xÞ�þ1

2Tr½γμSSðx;yÞγνSAðy;xÞ�:

ðC5Þ

Let us take the limitm ≪ T, wherem is the ψ mass, leadingin Fourier space to the following retarded self-energy of themediator:

Π00R ðPÞ ¼ −g2ψ8π

Zd4Kð2πÞ4 ½ðk

0 − p0Þk0 þ ðk − pÞ · k�½1 − 2nFðjk0jÞ�δðK2Þ 1

ðK − PÞ2 − isgnðk0 − p0Þϵ : ðC6Þ

Dropping the vacuum part and integrating over k0 one obtains:

Π00R ¼ g2ψ8π

Zd3kð2πÞ4

nFðjkjÞjkj

� ðjkj − p0Þjkj þ ðk − pÞ · kðjkj − p0Þ2 − ðk − pÞ2 − isgnðjkj − p0Þϵþ

ðjkj þ p0Þjkj þ ðk − pÞ · kðjkj þ p0Þ2 − ðk − pÞ2 þ isgnðjkj þ p0Þϵ

�:

ðC7Þ

The result so far is exact up to the fact that we neglectedthe ψ mass and ignored the vacuum contribution. Now, thehard-thermal-loop approximation [71] assumes that theexternal energy p0 and momentum jpj are smaller com-pared to the typical loop momentum jkj which is of theorder temperature (hard), since the integrand contains nF.Expanding the term in the brackets to leading order inp0=jkj and jpj=jkj, all remaining integrals can be per-formed analytically leading to the finite result:

Π00R;AðPÞ ≃ −m2

D

�1 −

p0

2jpj ln�p0 þ jpj � iϵp0 − jpj � iϵ

��; ðC8Þ

where the Debye mass is defined as m2D ¼ g2ψT2=3. This

result coincides with the result obtained from the imaginarytime formalism, where instead one has to perform a sumover Matsubara frequencies.


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APPENDIX D: VACUUM LIMIT OFTWO-PARTICLE SPECTRAL FUNCTION

Here we will derive Eq. (6.12) starting from Eq. (5.44) inthe limit of vanishing finite temperature corrections. As wehave briefly mentioned in Sec. V D, for the vacuum limitone has to carefully take into account the imaginary part iϵin the retarded equation, representing the small widthwhich will be taken to be zero in the end. We will seethat the result does not depend on this iϵ prescription aslong as ϵ is small enough.Suppose that the potential almost vanishes for a large

enough r. For a Yukawa type potential, this is true formVr ≫ 1. The ϵ parameter should be much smaller thanthis mass parameter, namely ϵ ≪ mV. In the case of theCoulomb potential one may introduce another small massparameter to the gauge boson. In the end of the computationone can take it to be zero while keeping mV ≫ ϵ. FormVr ≫ 1, the homogeneous solutions, g>;<, can be wellapproximated by the plane wave:

g>ðrÞ → C>eiffiffiffiffiffiffiME

pr−ϵr; ðD1Þ

g<ðrÞ →1

2

C<ei

ffiffiffiffiffiffiME

pr−ϵr þ C†

<e−iffiffiffiffiffiffiME

prþϵr

�: ðD2Þ

The Wronskian tells us that there exists a nontrivial relationbetween two coefficients C>;<. Since the Wronskian,WðrÞ ¼ g>ðrÞg0<ðrÞ − g0>ðrÞg<ðrÞ, does not depend on r,one may equate it at r ¼ 0 and r → ∞, which yields

1 ¼ Wð0Þ ¼ Wð∞Þ ¼ −iffiffiffiffiffiffiffiffiME

pC>C

†< ↔ C> ¼ iffiffiffiffiffiffiffiffi

MEp 1

C†<:

ðD3Þ

We have taken ϵ to be zero in the end of the computation.Let us evaluate the integral given in Eq. (5.44) by

means of Eq. (D2). The first observation is that thereis no imaginary part for g< if ϵ is zero from the beginn-ing. Thus, the integrand becomes relevant only afterr≳ ϵ−1 ≫ m−1

V . As a result one may evaluate the integralin Eq. (5.44) by substituting Eq. (D2):

Gρηξð0; 0;EÞjE>0;l¼0 ¼

1

2πTr½12×2�M

1

jC<j2limϵ→0

ℑ

×Z

∞

m−1V

dr1

cos2½ð ffiffiffiffiffiffiffiffiME

p þ iϵÞrþ δC<�

ðD4Þ

¼ 1

2πTr½12×2�M

1

jC<j21ffiffiffiffiffiffiffiffiME

p : ðD5Þ

Finally, substituting Eq. (D3) into this equation, wearrive at

Gρηξð0; 0;EÞjE>0;l¼0 ¼

1

4πTr½12×2�M2vreljC>j2; ðD6Þ

where we have used E ¼ Mv2rel=4.Now we are in a position to discuss its relation to the

conventional definition of the Sommerfeld enhancementfactor. In the limit of ϵ → 0, the wave function propagatesto infinity. Then one may obtain the Sommerfeld enhance-ment factor by extracting the amplitude of the wavefunction at the infinity, which is nothing but the relationSðvrelÞ ¼ jC>j2; see Ref. [44] for instance. Utilizing thisrelation, we finally get Eq. (6.12). For conventional reason,let us give the s-wave Sommerfeld enhancement factor forthe Coulomb case consistent with our equations:

SðvrelÞ ¼2παχvrel

1

1 − e−2παχvrel

: ðD7Þ

For the bound state, it is much easier to solve theequation directly rather starting from Eq. (5.44). One mayexpress the spectral function by means of the wavefunctions for the bound states [37]:

Gρηξð0; 0;EÞjE<0;l¼0 ¼ 2πTr½12×2�

Xn

δðE − EnBÞjψ ðBÞ

n ð0Þj2;

ðD8Þ

where ψ ðBÞn represents the normalized wave function for

the nth bound state. For instance, the wave function for thelowest energy state n ¼ 1 is given by

ψ ðBÞ1 ¼ 1ffiffiffi

πp

�αχM

2

�3=2

e−αχMr=2: ðD9Þ

The decay rate of the bound state is related to its wavefunction at the origin. For the lowest state, one can easilyshow this from Eq. (D9):

ðσvrelÞjψ ðBÞ1 ð0Þj2 ¼ 1

4Γ1S0 ; ðD10Þ

where the decay rate of the lowest bound state is given by

Γ1S0 ¼ ðα2χ þ αχαψÞα3χM2: ðD11Þ

Similar calculation holds in the limit of negligiblethermal width but finite real-part corrections. For thiscase one should substitute the kinetic energy E → E −ℜVeffð∞Þ ¼ Mv2rel=4 and similar for the bound-stateenergy. Also one has to take ϵ smaller than mD and mV .


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APPENDIX E: NUMBER DENSITY ANDCHEMICAL POTENTIAL IN GRAND

CANONICAL ENSEMBLE

In this section, we present an alternative way of howto derive the chemical potential as a function of the totalnumber density directly from the EoM. It is convenient towrite the EoM in integral form as

Gηðx; yÞ ¼ Gη;0ðx; yÞ − g2χ

Zw;z∈C

Gη;0ðx; wÞDðw; zÞ

× ½Gηξðw; z; y; zÞ −Gηηðw; z; y; zÞ�; ðE1Þ

Gξðx; yÞ ¼ Gξ;0ðx; yÞ − g2χ

Zw;z∈C

Gξ;0ðx; wÞDðw; zÞ

× ½Gηξðz; w; z; yÞ −Gξξðw; z; y; zÞ�: ðE2Þ

The number density is given by theþ− component, namelyin Wigner coordinates it reads nηðR; TÞ ¼ Tr½Gþ−

η ðx; xÞ� ¼Tr½Gþ−

η ð0; 0;R; TÞ�. Now one can already clearly see thestructure that the total number density is given by theideal number density plus interactions. These equations areexact within our nonrelativistic effective action. ExpandingD around the narrow width limit and assuming a grandcanonical state, one should recover Eq. (6.35) for thenumber density since the only assumption entering there isthe narrow width approximation. Equation (6.35) is thecorrect thermodynamic definition of the number densitywhich should coincide with the result obtained by solvingfor the number density from these integral EoM whenassuming a thermodynamic picture like the grand canonicalensemble. It requires a rigorous proof of this claim, whichwe would like to give somewhere else.Instead, we would like to give some approximations

in order to obtain an analytic solution of Eq. (E1). Werestrict the discussion by assuming the system is in a grandcanonical state. Utilizing the KMS condition for finitechemical potential, one can formally solve Gþ−

η in terms ofspectral function:

nη ¼ Tr½Gþ−η ð0; 0Þ�

¼Z

d3pð2πÞ3

dωð2πÞ

1

eβðMþω−μηÞ þ 1Tr½Gρ

ηðp;ωÞ�

≃Z

d3pð2πÞ3

dωð2πÞ e

−βðMþω−μηÞTr½Gρηðp;ωÞ�: ðE3Þ

In the last equality we assumed the DM gas to be diluteand approximated the Fermi-Dirac distribution as aMaxwellian. By KMS relation we have formally solvedfor the number density in terms of chemical potential andspectral function. The spectral function can be computedfrom the retarded componentGR

η , according toGρη;0ðp;ωÞ¼

GRη;0ðp;ωÞ−GA

η;0ðp;ωÞ (see Sec. II). The EoM of the

retarded correlator can be obtained from Eq. (E1) bysubtracting þ− from the þþ component, e.g., GR

η ¼Gþþ

η −Gþ−η . In subsequent sections we solve the retarded

equation in various approximations, compute the spectralfunction and finally evaluate Eq. (E3).

1. Ideal gas approximation

The ideal gas approximation can be defined as thezeroth order contribution in Eq. (E1). This means we haveto know the solution of the free retarded correlator. Itcan be obtained from the differential form of the retardedequations. In Fourier space of the microscopic Wignercoordinates it is given by GR

η;0ðp;ωÞ¼ iδij=ðω−p2=ð2MÞþ iϵÞ, leading to Gρ

η;0ðp;ωÞ¼δijð2πÞδðω−p2=2MÞ.Inserting this into Eq. (E3) results in the ideal gasapproximation of the free number density:

nη;0 ¼ 2

Zd3pð2πÞ3 e

−βðMþp2

2M−μηÞ: ðE4Þ

We can invert this relation to obtain the chemical potentialas a function of the number density in the ideal gasapproximation:

βμidη ¼ ln

�nη;0neqη;0

�; where neqη;0 ¼ 2

Zd3pð2πÞ3 e

−βðMþp2=2MÞ:

ðE5Þ

And similar expressions for antiparticle ξ.

2. Hartree-Fock approximation

The Hartree-Fock approximation is the zero orderapproximation of the four-point correlator. UsingEqs. (4.9)–(4.10), one obtains

½Gηξðx; z; y; zÞ −Gηηðx; z; y; zÞ� ≃Gη;0ðx; zÞGη;0ðz; yÞ;ðE6Þ

½Gηξðz; x; z; yÞ −Gξξðx; z; y; zÞ� ≃ Gξ;0ðx; zÞGξ;0ðz; yÞ;ðE7Þ

valid for symmetric DM. Inserting this into Eqs. (E1)–(E2)and subtracting the components Gþþ

η − Gþ−η , one obtains

for the retarded equations:

GRη ðx; yÞ ¼ GR

η;0ðx; yÞ þZ

d4zd4wGRη;0ðx; zÞð−iÞ

× ΣRη ðz; wÞGR

η;0ðw; yÞ; ðE8Þ

GRξ ðx; yÞ ¼ GR

ξ;0ðx; yÞ þZ

d4zd4wGRξ;0ðx; zÞð−iÞ

× ΣRξ ðz; wÞGR

ξ;0ðw; yÞ: ðE9Þ


115023-39

The single particle self-energies are defined on the CTPcontour as we have introduced in Eq. (5.20). The retardedcomponent is ΣR ¼ Σþþ − Σþ− and can be written as

ΣRðx; yÞ=ð−ig2χÞ¼ Dþþðx; yÞGþþ

0 ðx; yÞ −Dþ−ðx; yÞGþ−0 ðx; yÞ ðE10Þ

¼ Dþþðx; yÞGR0 ðx; yÞ −DRðx; yÞGþ−

0 ðx; yÞ ðE11Þ

¼ D−þðx; yÞGR0 ðx; yÞ −DRðx; yÞGþ−

0 ðx; yÞ: ðE12Þ

In the last step, the definition Dþþðt; t0Þ ¼ θðt − t0ÞD−þðt; t0Þ þ θðt0 − tÞDþ−ðt; t0Þ was used and the fact thatthe retarded function GR

0 projects out only the D−þðt; t0Þcontribution due to equal times in D and GR

0 . A keyobservation is that the self-energy can depend on the DMnumber density and chemical potential due to the Gþ−

0

contribution. This might lead to a nonlinear dependence ofthe total number density on eβμ when inserting the spectralfunction obtained from the retarded equation into Eq. (E3).In the following we would like to perturbatively resumEqs. (E8) and (E9) which brings us later to the widely usedSalpeter correction. Let us therefore drop the dependenceof the self-energy onGþ−

0 ðx; yÞ. Then the equation is closedin terms of the retarded correlators GR

0 . Within thisapproximation, the self-energy can be written in Fourierspace as (to leading order gradient expansion):

ΣRη ðp;ωÞ

¼ g2χ

Zd3p1

ð2πÞ3dω1

ð2πÞdω2

ð2πÞGρ

η;0ðω1;p − p1ÞD−þðω2;p1Þω − ω1 − ω2 þ iϵ

ðE13Þ

¼ g2χ

Zd3p1

ð2πÞ3dω1

ð2πÞD−þðω1;p1ÞΩ − ω1 þ iϵ

ðE14Þ

¼ g2χ1

2

Zd3p1

ð2πÞ3dω1

ð2πÞD−þðω1;p1Þ þD−þðω1;p1Þ

Ω − ω1 þ iϵ

ðE15Þ

¼ g2χ1

2

Zd3p1

ð2πÞ3dω1

ð2πÞD−þðω1;p1Þ þDþ−ð−ω1;p1Þ

Ω − ω1 þ iϵ

ðE16Þ

¼ −ig2χ1

2

Zd3p1

ð2πÞ3DþþðΩ;p1Þ; ðE17Þ

where Ω ¼ ω − ðp − p1Þ2=2M. The final form is conven-ient for inserting the static HTL approximation of Dþþ asgiven in Eq. (3.11). Performing the one-loop calculationresults in

limΩ→0

ΣRðp;ωÞ ¼ −ig2χ1

2

Zd3p1

ð2πÞ3 limΩ→0DþþðΩ;p1Þ

¼ −1

2ðαχmD þ iαχTÞ: ðE18Þ

One can recognize that this result is exactly half theeffective in-medium potential for two particles at largedistance; see Eq. (5.32). Now, for perturbative resummationwe replace the retarded correlators at the end of Eqs. (E8)and (E9) by the fully dressed one: GR

η;0ðw; yÞ → GRη ðw; yÞ,

GRξ;0ðw; yÞ → GR

ξ ðw; yÞ. Performing Wigner and Fouriertransformation of the equation leads at the leading orderin gradient expansion to

GRη ðp;ωÞ ¼ GR

η;0ðp;ωÞ þGRη;0ðp;ωÞð−iÞΣR

η ðp;ωÞGRη ðp;ωÞ;ðE19Þ

and similar equation for the antiparticle. Then, by usinggeometric series one ends up with the HTL single particlecorrelators:

GRη ðp;ωÞ ¼

iδijω − p2=2M − ΣR

η ðp;ωÞ þ iϵ;

GRξ ðp;ωÞ ¼

iδijω − p2=2M − ΣR

ξ ðp;ωÞ þ iϵ: ðE20Þ

Computing the spectral function from the difference ofretarded and advanced correlators results in a Breit-Wignershape:

Gρηðp;ωÞ¼ δij

Γη

ðω−p2=2M−ℜðΣRη ÞÞ2þðΓη=2Þ2

; ðE21Þ

where the particle width is defined by Γ≡ 2ℑðΣRÞ. Insummary, we evaluated the four-point correlator in theHartree-Fock (HF) approximation and formally solvedfor ρ as a function of the self-energy in static HTLapproximation, shifting the energy by αχmD=2 and broad-ening the peak via imaginary contributions. Finally, let usquote the chemical potential in HF and static HTLapproximation:

βμHFη ¼ ln

�nη

neqη;HF

�; where

neqη;HF ¼ 2

Zd3pð2πÞ3

dωð2πÞ e

−βðMþωÞ

×Γ

ðω − p2=2M −ℜðΣRη ÞÞ2 þ ðΓ=2Þ2 : ðE22Þ

This approximation might be already good enough if thereare no bound states but the two-particle spectral functionhas support at negative energies due to thermal width. Wealso see that one-particle spectral function can have spectralsupport at negative energies.


115023-40

3. Salpeter correction

Taking the limit Γ → 0 in the spectral function Eq. (E21)results in

limΓ→0

Gρηðp;ωÞ ¼ δijð2πÞδðω − p2=2M −ℜðΣR

η ÞÞ: ðE23Þ

This is called the narrow width or quasiparticle approxi-mation, taking only the real-part correction into account.Inserting the spectral function into Eq. (E3) leads to thechemical potential and equilibrium number density:

βμSPη ¼ ln

�nηneqη;SP

�; where

neqη;SP ¼ 2

Zd3pð2πÞ3 e

−βðMþp2=2Mþℜ½ΣRη �Þ: ðE24Þ

For the self-energy in the static HTL approximation, thereal-part correction is ℜðΣR

η Þ ¼ −αχmD=2, according toEq. (E18). This is the well-known Salpeter correction tothe equilibrium distribution.The Salpeter correction is a simple first order approxi-

mation for the description of quasi particles in a plasma.As we have seen in Sec. VI D, it might be however requiredfor our number density equation to compute both, theannihilation/decay rates and the chemical potential, to thesame level of approximation in the four-point correlator.Especially when bound-state solutions exist it is requiredto solve the four-point correlator nonperturbatively (byresummation). The Salpeter correction was obtained byapproximating the four-point correlator as a product of freeparticles without self-interactions; see Eqs. (E6) and (E8).Thus, in this approximation of the chemical potential,bound-state contributions never appear. It might lead to

inconsistencies in the number density equation, like expo-nentially growing terms for late times.Let us illustrate why the Salpeter correction is not enough

to correctly describe the freeze-out at late times. We plug thechemical potential μSPη in Salpeter approximation into ourmaster formula for the number density and obtain

_nη þ 3Hnη ¼ −2ðσvrelÞGþþ−−ηξ;s ðx; x; x; xÞjeq½eβ2μSPη − 1�

ðE25Þ

¼ −2ðσvrelÞGþþ−−

ηξ;s ðx; x; x; xÞjeqneqη;SPðTÞneqη;SPðTÞ

½nηnη − neqη;SPðTÞneqη;SPðTÞ�:

ðE26ÞThere are two reasons why this description fails atlate times if bound-state solutions exist. First of all,since no bound states are included in the computationof the chemical potential in Salpeter approximation, theionization degree would always be approximated as 1.Second, if one would compute the spectral function inGþþ−−

ηξ;s ðx; x; x; xÞjeq nonperturbatively, this would cause anexponential growing term at late times, caused by the boundstates:Gþþ−−

ηξ;s ðx; x; x; xÞjeq ∝ eβjEBj. In the Salpeter approxi-mation, the denominator n2η;SP cannot kill this unphysicalbehavior. Note however, if one computes the chemicalpotential to the same level as the annihilation rates, thedegree of ionization exhibits a term leading to a cancellationof the exponential growing term in Gþþ−−

ηξ;s . Our mainnumber density equation automatically incorporates thisbecause it evaluates the chemical potential and numberdensity to the same level of approximation. The Salpetercorrection together with solving the spectral function non-perturbatively was used in Refs. [55–57].

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